And the other way around - are we allowed to train and use neural nets on anyone's work, assuming no near duplicates are generated?

Technically the neural net does not have a copy of the originals inside, it has their summed up gradients. The original model shrunk 4B images to a mere 4GB file, so 1 byte/image worth of information learned from each example, a measly pixel.

The model has learned un-copyrightable things like the shapes of various objects and animals, how they relate, their colours and textures. That should be protected. Let them finally learn how many fingers we have.