Is there a good notion of `Separated Stack'?

Look at Def. 4.7 of Deligne--Mumford for the definition when $X$ is DM: they define $X$ to be separated if $X \to X \times X$ is proper (or equivalently, finite).

One can first define a 'proper algebraic space' $X,$ using its 'underlying space' $|X|,$ and then define a morphism of algebraic spaces $f: X \to Y$ to be proper if for any affine (or just quasi-compact) scheme mapping into $Y,$ the fiber product gives a proper algebraic space. Finally define an Artin stack $X$ to be separated if the diagonal (which is representable) is proper, i.e. for any algebraic space mapping into $X \times X,$ the fiber product....

One can also define a 'proper Artin stack' similarly. See Laumon and Moret-Bailly.