Idioglossia is known among a very small number of identical twins, most infamous in a handful of cases where they use this to cut themselves off from the rest of the world, retreating into a kind of self-imposed linguistic prison - keeping others out, locking themselves in - which can then be used, in extreme cases, to commit crime.

I'm a little skeptical of some of the things on this page: (as it's sensationalist, to sell a crime book), but ... "Identical twins Jeff and Greg Henry were close as brothers could be, inventing their own language and often exchanging identities." There was another case (which I can't find on the web) of British identicial twin girls who invented a private language, refused to speak to even their own parents, murdered someone, and refused to communicate with any representatives of the law in their defence (I think I remember that accurately). A TV programme was made about them some years ago.

This page, on the case of identical twins Poto and Cabengo, has their father forbidding them to use their private language, as he wants them to become a part of society, not self-excluded from it. (Their language turns out not to be thoroughly original, though, but a mixture of mashed-up English and German - their parents were American and German.)

Which raises a related point. If one person creates a syntax, grammar, etc. for a language, and only that one person speaks it, is it really a language at all? Ludwig Wittgenstein thought not, and I'm inclined to agree. Words reach their meaning through the context in which they are used, and a sound only becomes meaningful by being used in community, by commuunicating something meaningful. If I say to you, "Wera con bogora sgog, puta monh jhytrew", have I communicated anything? Is there a community of people who have collectively and over time understood what these words should sound like when spoken and what they mean? No, of course not, I just made it all up, so it cannot be a language, since nothing is communicated, even if I privately decide what it means myself and develop it over a number of years.

Two people might just about be said to have a private language, if two people can be privately understood by each other. But one?