So, the only hunch I have is that there might be a kind of cryptographic/information approach to it. If you treat natural language as a noisy encoding of some underlying set of intents (whatever sentience is), then the question becomes "is there enough Information in the signal (natural language) to accurately model the underlying intents statistically?". If the answer is 'no', or 'yes, but very badly', then it seems likely to me that there's some shared hardware/firmware (in this framing, the functional equivalent of a decryption key) between people. If the answer is 'yes', or 'yes, mostly at least', then there's no need for any kind of universal grammar or such, and you can learn language from a blank state. Basically, is natural language something like a one-time pad, or is it more like a breakable form of encryption?

Of course, there are plenty of assumptions in that model (for one, can you even separate 'language' from 'sentience' like an information model would suggest. I'm not so sure you can, but then I've always been somewhat sympathetic to Sapir-Whorf). I'm also not a competent enough mathematician in these fields to see where all the holes in this line of thinking are.