If $X$ is an algebraic stack fibered in sets (and therefore essentially a sheaf), is it an algebraic space? It seems conceivable that at least when $X$ is Deligne-Mumford, it is actually an algebraic space. However, when $X$ is an Artin stack, it is only required to have an atlas of smooth maps of affines, so it seems less likely that this would be the case. 



Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?
Yes. The criterion for an Artin stack to be Deligne-Mumford is that it should have unramified diagonal (this is somewhere in Laumon Moret Bailly, I don't have it here). If the stack is fibered in sets, the diagonal is a monomorphism, and a monomorphism is certainly unramified.


