Puzzle me this

Trump administration fires top copyright official days after firing Librarian of Congress

Perlmutter’s office recently released a report examining whether artificial intelligence companies can use copyrighted materials to “train” their AI systems and then compete in the same market as the human-made works they were trained on.

In today's world, Perlmutter's dismissal isn't much of a puzzle — she has a foreign-sounding name and the wrong color hair.

But this one: If an AI is "trained" on copyrighted works do the original copyrights still apply to the AI-generated product?

Or, even more so, this one: Who owns the rights to an AI-generated document? 

And, if I pay a subscription price for the using the AI, can I claim the rights to its output as a work for hire?

O brave new world.