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For lions to understand each other, what they talk about must be perceptible. Even if lions talk about the special experiences of having a tail, they can only do so because tails are visible to them. If it is visible to them, it will be for us, so anything they talked about, we could also soon understand. (The communicational gulf between individuals is greater than that between types of individual.)
But consider bees: they see infra-red … so they do live in a different visible world. Would that not separate them from us? No, even that cannot obstruct language. At first we would be very puzzled, and not understand them. But after a while we would realise that they are seeing something we do not. We understand what ‘seeing’ means in terms of its influence on behaviour, so we can deduce from that the missing piece that we cannot see. So, it is remarkable that even when we cannot see something we can still in a way see it ‒ we can see its indirect effects on and relations with other visible phenomena. Even what we cannot describe, we can still describe. But really this is how all description works: things are only ever known by how they affect other things.
‘What does infra-red mean?’ That is at first obscure, but think like this: seeing in infra-red only means something in terms of seeing not in infra-red. Part is in common and part not. Everything means something only by referring to something else: everything is graspable by being similar to something else we already grasp. Language is, at scale, an unbroken network: nothing is isolated and unreachable ‒ just as the physical world is all connected.
So animals cannot have a private language any more than particular people can. The only things we could not understand are those that do not exist for us ‒ and then we are missing nothing at all.
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To clarify the relation of human and lion, it seems better to look from the other direction. We could understand lions, but they could not understand us ‒ that asymmetry is informative. They lack a sufficient sophistication of pattern handling (one might say, intelligence). The issue here is not language/data, but algo/processing: it is not a matter of transfer of data, it is what one can to do with it. Lions can see what we see, but they cannot follow the more complex patterns we do.
Furthermore, in principle, every language is entirely translatable, and all aliens are completely understandable. (Assuming sufficient evidence of the subject matter, of the world, the language is talking about.) Because understanding is modelling, and everything can be modelled with a turing machine. The only question is how accurately and how efficiently. An alien might have better algos and execution speed that we could not keep up with at current knowledge.
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… Nonetheless (as a purely empirical matter): if a red herring could talk, we should not believe it.
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