There are reports of twins bootstrapping off each other, from the principle of noise->action->repeatnoise, called idioglossia. Seems is not that great actually as language. This NYT blog post suggests the words are more babble than language, which matches how my daughter spoke to the cat: English intonation and facial expressions, meowy babble as words. The Wikipedia article on cryptophasia says "While sources claim that twins and children from multiple births develop this ability perhaps because of more interpersonal communication between themselves than with the parents, there is inadequate scientific proof to verify these claims."

I've heard anecdotes about things like children spontaneously developing their own languages even when completely deprived of language in their environment, which would weakly indicate the contrary position. Unfortunately, I don't know whether to trust said anecdotes -- can anyone corroborate?

There are examples of groups of deaf people developing languages together, but generally over a generation or two, and in large groups. The most prominent such case is Nicaraguan sign language.