For super permissible licenses like MIT then it's probably fine. Maybe folks would need to list the training data and all the licenses (since a common requirement of many of even the most permissible licenses is to include a copy of the license).

As far as I know, a court hasn't ruled on whether clauses like "share alike" or "copy left" (think CC BY-SA or GPL) would require anything special or not allow models. Anyone saying otherwise is just making a best guess. My best guess is (pessimistically) that it won't do any good because things produced by a machine cannot be copyrighted. But I haven't done much of a deep dive. I got really interested in the differences between many software licenses a few years back and did some reading but I'm far from an expert.