Firstly, Wittgenstein makes a clear distinction between reality and dreams; or between language use in reality and language "use" in dreams (where the latter isn't really use):

I cannot seriously suppose that I am at this moment dreaming. Someone who, dreaming says: “I am dreaming,” even if he speaks audibly in doing so, is no more right than if he said in his dream “it is raining,” while it was in fact raining. — On Certainty 676

Secondly, even if we suppose in your scenario that you can "talk to people in both worlds in different languages" then you must be talking to someone else in a public language that you and the people you are conversing with share (albeit in your dreams). Otherwise, you are not really talking to anyone else at all.

The point is that a private language is impossible. Therefore, as you say, "nobody in existence could call any language private".

I cannot seriously suppose that I am at this moment dreaming. Someone who, dreaming says: “I am dreaming,” even if he speaks audibly in doing so, is no more right than if he said in his dream “it is raining,” while it was in fact raining. — On Certainty 676

Meh, I've realized I was dreaming before and recall even saying so "out loud" to a a dream person.

Secondly, even if we suppose in your scenario that you can "talk to people in both worlds in different languages" then you must be talking to someone else in a public language that you and the people you are conversing with share (albeit in your dreams). Otherwise, you are not really talking to anyone else at all. — Luke

That would be the position of the solipsist. So the question is whether solipsism can be defeated by saying the solipsist is not really talking to anyone. Which is true, but the solipsist can just say they have an experience of talking to people, just like in a dream, and that's all it is.

The point is that a private language is impossible. Therefore, as you say, "nobody in existence could call any language private". — Luke

Is it, though? A BIV has experiences of language with other people, but that language experience is fed to them by the vat. For the impossibility claim to work, you have to assume other people exist. It's not a defeater if you're willing to engage in radical skepticism.

Meh, I've realized I was dreaming before and recall even saying so "out loud" to a a dream person. — Marchesk

Your awareness of dreaming is not at all the same as Wittgenstein's inability to seriously suppose that he is dreaming while he is awake writing his book. Furthermore, your comments have little relevance to the current discussion of private language. If we distinguish talking from talking-in a-dream, Wittgenstein is discussing the former whereas you are discussing the latter.

Solipsism or not, if we suppose that you are talking-in-a-dream to other people in a language that is understood by your dream participants, then it is not a private language. But that's a lot to suppose, and I'd be unwilling to consider a dream-language a real language. Unless, perhaps, you could actually teach it to someone else in waking reality.

As for BIVs, do you know any?

Solipsism or not, if we suppose that you are talking-in-a-dream to other people in a language that is understood by your dream participants, then it is not a private language. — Luke

It is, because the dream participants are me. Or to be more accurate, the dream participants are experiences only, not other people. A solipsist engages in private language by definition, so solipsism would have to be ruled out for a private language to be impossible.

As for BIVs, do you know any? — Luke

This is a philosophical discussion, and if someone makes that claim that something is impossible, then it's right to point out scenarios where it's not. Impossibility is a very strong claim.

In a dream or in solipsism I do not own or experience another person's consciousness with whom I'm conversing. It maybe imaginary, but I have no control over another person's consciousness, and I don't experience it. While I experience my own.

This may be taken that the imaginary or dreamt person's consciousness either does not exist, or else it is a function of mine, but it is not under the control of mine.

This is difficult to imagine, that I don't have a control over some function that I fully own. But it is the bread-and-butter for solipsism.

Therefore private language does not exist in solipsism, even if the only person who speaks it or understands it is myself, that is, only one person. Because the language I use in dreams or in solipsism still communicates with an entity that is not under my control, therefore on some level it's a separate entity from me. And therefore language is practiced by more than one entity.

I think the most coherent version of solipsism is just experiences with no self owning anything. Self is just another experience of the single stream, which is all there is.

Therefore there is private language existing in solipsism. Thank you for disproving and/or proving your own thesis. (I forgot whose side you are on... I can't go back and look when I compose a reply. Sorry.)

@Luke

Solipsism or not, if we suppose that you are talking-in-a-dream to other people in a language that is understood by your dream participants, then it is not a private language.

I'm rather surprised by this statement. So talking to people I'm imagining in a language that can not be taught to anyone real would not be considered a private language by you?
This would make things a lot more sensible for me, however very uninteresting too.
Because now every language or sign is obviously public now, as I can simply imagine (that's an ok substitute for dreaming right?) somebody to talk to in this language.
I find it hard to believe that people have come to argue about anything, assuming they agree on this definition of a private language.

Now this person can talk to people in both worlds in different languages and I believe it makes sense to say that from their perspective all of these languages are public. — SomeName

That part doesn't follow from anything prior to it, and if you're trying to argue that language isn't private, you can't just jump to the conclusion you're shooting for in the middle of the argument.

For a person from one of these worlds, that can only experience one of them and not travel between them, the languages from the other world are completely unintelligible and they would simply not call it a language. — SomeName

I don't think this part works, either. For one, we don't say that someone is speaking a language or not based on whether we understand what they're saying. We say that someone is speaking (or writing) a language if it seems to function as a language to them--it's used to communicate, to record information, etc. As it is, there are a handful of ancient languages that we haven't been able to crack yet, and maybe we'll never crack them. We don't say that they're not languages because of this.

Secondly, folks who posit that language is private, or at least has some private aspects--I'm one of them--obviously don't say that this makes language unintelligible.

But as we can at any point be only one person and have only one point of view we are unable to transcend into some meta view where we could call a language private relative to somebody else. — SomeName

That part I'm not sure I understand. Could you explain it in other words?

Re the more strict Wittgensteinian idea of whether there could be a private language that turns out to be indecipherable to others in perpetuity (I wouldn't say "in principle indecipherable" because I don't think it really makes sense to talk about principles in this context), I think that's possible, but usually this just devolves into a rather facile disagreement about how folks are pledging to use the term "language."

My understanding of the "Private Language Argument" is that it would be impossible for a lone individual to create a language if they did not already speak a public language, and that if they did create a language they would need to translate their new words for things, actions, qualities and so on into their native language equivalents in order to be able to specify to themselves what they mean by them. But this would mean that their language is not genuinely private, because it is understandable only in terms of an existing public language.

That part doesn't follow from anything prior to it, and if you're trying to argue that language isn't private, you can't just jump to the conclusion you're shooting for in the middle of the argument. — Terrapin Station

You're correct, this is still part of the set up.

I don't think this part works, either. For one, we don't say that someone is speaking a language or not based on whether we understand what they're saying. We say that someone is speaking (or writing) a language if it seems to function as a language to them--it's used to communicate, to record information, etc. As it is, there are a handful of ancient languages that we haven't been able to crack yet, and maybe we'll never crack them. We don't say that they're not languages because of this. — Terrapin Station

Please substitute "impossible to learn" for unintelligible. That's what I meant.

That part I'm not sure I understand. Could you explain it in other words? — Terrapin Station

The point of my set up was to exemplify the relative nature of language.
To one person it's intelligible, while it maybe completely impossible to learn for the other,
since they do not share any experiences (except for that one person living in both worlds, so that's a weak point of the argument).
Two worlds A,B and people a,x,b. a lives n A, b lives in B, x lives in both.
In A and B different languages are spoken, also A and B do not share anything but x.
The meta view is exactly this formalization of the two worlds. In this meta view we understand that from the point of view of a, x does speak weird gibberish, but we also know that this gibberish is indeed an actual language. One could say the language spoken in B is a private language in A used only by x.
However as we can choose only the viewpoint of a,b or x in real life, there are only these possibilities:
If we choose a: x is talking gibberish, it's not a language.
If we choose x: I know all languages, they all are public, as I talk to a and b.
If we choose b: same as a.
So there is no private language here, even with the more permissive use of the term above.

To repeat my alleged relation to Wittgensteins argument:
Wittgenstein talks about someone relating a sign to an experience privately,
here I'm replacing the single sign with a complete language spoken to people, that
are either imagined or simply not considered their own people, at least from the point of view of a or b.
Thus x's experience of talking to people "from the other world" is a private one from the point of view of a or b. I do this because I do not consider it sensible to divide people into real und unreal and then consider talk to unreal people private, as from any point of view the unreal people simply do not exist.
The distinction exists only in a "meta-view" which is not clearly connected to the actual problem, but instead a game of words, played by rules which maybe do not bear any connection to the actual life of a,b or x.

My understanding of the "Private Language Argument" is that it would be impossible for a lone individual to create a language if they did not already speak a public language, and that if they did create a language they would need to translate their new words for things, actions, qualities and so on into their native language equivalents in order to be able to specify to themselves what they mean by them. But this would mean that their language is not genuinely private, because it is understandable only in terms of an existing public language. — Janus

This would work for me, if I would not believe it to be possible for someone to imagine a whole world. But maybe exactly that is my mistake. In this made up world, there would be actions and things that one could talk about, you see? But again, maybe one can not really make up things.

they experience while being awake. These two worlds this person lives in do not share anything, neither sensations nor rules. Now this person can talk to people in both worlds in different languages and I believe it makes sense to say that from their perspective all of these languages are public. — SomeName

Once there's a community, whether in dreams or reality, there is no private language.

As defined, a private language has only ONE user.

If wikipedia is correct then Wittgenstein's argument hinges on the undefinable status of entities in a private language. The only definitional method available in a private language is the ostensive technique and that according to him is so flexible that almost any meaning can be derived from it and so it's unreliable. Also, according to Wittgenstein, it becomes impossible to check if the signs are being used correctly or not for the above reasons.

Two worlds A,B and people a,x,b. a lives n A, b lives in B, x lives in both.
In A and B different languages are spoken, also A and B do not share anything but x.
The meta view is exactly this formalization of the two worlds. In this meta view we understand that from the point of view of a, x does speak weird gibberish, but we also know that this gibberish is indeed an actual language. One could say the language spoken in B is a private language in A used only by x.
However as we can choose only the viewpoint of a,b or x in real life, there are only these possibilities:
If we choose a: x is talking gibberish, it's not a language.
If we choose x: I know all languages, they all are public, as I talk to a and b.
If we choose b: same as a.
So there is no private language here, even with the more permissive use of the term above. — SomeName

This sounds like my comment above: "usually this just devolves into a rather facile disagreement about how folks are pledging to use the term 'language.'"

So in other words, the person is making a pledge to not use the term "language" when they think the language is gibberish.