To take that a step further, why couldn't such an AI periodically generate questions to ask itself?

Being good at generating questions, could become almost as important as answering them.

Would it be posible to run two instances of it and have them talk to each other?

Yes, they could have multiple instances, go back and forth with each other. This strategy was used with various chess AI in order to help improve it.

However, to make it more human-like, I would think they would make some type of AI self reflection. So like we can talk and think out problems to ourselves, perhaps they would have the AI periodically practice questioning itself, when humans are not around or asking it questions, and do some form of self-analysis.

The training corpus is atemporal so this would be either meaningless or equivalent to a random prompt.

A chatbot with a sense of time learned by also training with response rates, complete editing sessions, some formalization of interlocution, etc. would be interesting, but that's not what these are.