Many people have asked this question. I am not aware of any definitive, study-based answer. However, anecdotally you will find that people who compost stuff containing DE do not generally feel like they're seeing obvious effects. Personally I would not expect much if any effect, insofar as most of what DE does, it does only in a dry environment, whereas a compost pile (and the soil you use the compost in) are at least moderately damp.
Still, there is no definitive empirical answer yet that I know of, so it's up to you what you want to do. Certainly the less DE you use in your coop, the less likelihood of a problem with it in the compost.
(e.t.a. -- it will never be "gone", though. DE does not break down or decompose into other substances, not meaningfully. It is silicon, ideally in amorphous rather than crystalline form. It will not transmute to another element <g> and it is really quite nonreactive. Think of it as sand. Sand does not go away, it just gets smaller and smaller, and may be diluted out by other stuff that is mixed in)
Personally I think an awful lot of people on this forum throw around WAY too much DE for not very compellingly good reasons, sometimes to the point of respiratory hazard, more often just in a wasteful way. But, it's a free country.
Good luck, have fun,
Pat