The conditional distribution is the same. The joint distribution of the full output does change with watermarking. Just look at the diagram in the article: there is a loop back to the seed once a token has been generated.
Changing the distribution is the whole point: it introduces statistical regularities that can be detected.
It introduces statistical regularities, but all RNGs introduce statistical regularities. So the question is on average are these statistical regularities better or worse than those introduced by the alternative, and the answer is no, if implemented properly.
> It introduces statistical regularities, but all RNGs introduce statistical regularities.
Agreed.
> So the question is on average are these statistical regularities better or worse than those introduced by the alternative,
Indeed. This is an empirical question. That is what the article is about, for a specific setting.
> ... and the answer is no, if implemented properly.
I don't agree with that. This is not about implementation. It's about how the statistical regularities that are imposed on the full output distribution affect that distribution. There is a change - by construction. That change can be good in some situations and bad in others. The authors claim that it is mostly bad in the setting that they investigated. This looks like a fair statement.
Changing the distribution is the whole point: it introduces statistical regularities that can be detected.