One is when copyrighted material is "pirated" for use in training, i.e. you torrent "the pile" instead of paying to acquire the books.
The other is when a user uses an LLM to generate a work that violates copyright.
Training itself isn't a violation, that's common sense. I am aware of lots of copyrighted things, and I could generate a work that violates copyright. My knowing this in and of itself isn't a violation.
The fact that an LLM agrees to help someone violate copyright is a failure mode, on par with telling them how to make meth or whatever other things their creators don't want them doing. There's a good argument for hardening them against requests to generate copyrighted content, and this already happens.