But why impossible? Take this example: https://en.wikipedia.org/wiki/Zero-knowledge_proof#Discrete_... - all parties exchange are essentially random numbers (which doesn't count as "additional information"), yet they establish the fact that Peggy knows x. There's that footnote about how r must be truly random, of course, but as long as everything's done right there is no additional information there whatsoever, only random noise.

Side note: one thing I certainly don't understand are NI-ZK - while I think I have some rough understanding of how interactive ZK proofs work - well, I mean, I've read Quisquater et al.'s "How to explain zero-knowledge protocols to your children" - but I'm pretty clueless how they manage to make such things non-interactive.

I suppose the fact that it's Peggy and Victor who are running that protocol is simply irrelevant. They're talking about that value of x after all, and nothing related to that topic is revealed beyond the proof of the fact that Peggy knows it. If she'd appear in person, wherever she wears a red shirt or a white one would be might be considered "information" but it would be completely out of scope.