G: Q -> (X,Y)
where Q is some prior from which you are sampling. If they are not invertible then you straight up cannot get P(X,Y) out of the generator. Even if it is invertible getting P(X) requires integrating out the Y which might be infeasible (since the model is not integrable and is sufficiently fast changing that you need very, very many samples).
Say you use regular expressions to extract sensitive data from standardized, but numerously varied, form documents. The pieces of information extracted are very common classes of data: first name, last name, dates, physical locations.
During the extraction process you can save the complement of the extraction (the "leftovers") and insert generated data at the extraction points. Also, because you've extracted the actual sensitive data, you can exclude that from the set of values used for generation, if it's practical.
Sometimes people get caught up in the math and theory that they fail to see the practical solutions.