Of course, once the AI is trained, you can’t look at some arbitrary output and determine whether that specific output came due to some specific training data set. In principle, if some of your training data is found to violate copyrights you either have to compensate the copyright holder or re-train the model without that data set.
Finding out whether a copyrighted work is part of the training data is a matter of going through it, and should be the responsibility of the people training the model. I would like to see a case where it has been shown that a copyrighted dataset has been used to train a model, and those violating the copyright by doing so are held responsible.