The copyright, if any, would not reside with the celebrity unless it had been assigned to them by the author of the document. An odd thing about lawyers is that they do not claim copyright in a lot of the things they write, such as contracts, briefs, etc., but I see no reason why they couldn't.
I would peg the probability of "running into problems" as low, just off the top of my head, and would be interested in knowing what some of the other lawyers here think. This is not a legal opinion on the subject.