I think a reasonable thing folks want is to not have their art or design used to train these models without permission.
Unless they are saying they don't want anyone "trained" on "their data", it's a phrase that simply makes no sense and only expressed by people who don't know how the real world works at all.
spare me the false equivalence of pattern theft from human beings learning or reacting.
the express purpose of said pattern ripoff machines is to obviate me and regurgitate likenesses of my work, so, no, i shall not participate in my own murder.