When we study sheaves of sets (on a space X or a site C) we are often interested in the stalks of the sheaf (at either a point $p:1\to X$ or a left exact, cover-preserving functor $a:C\to Sets$). I wonder how one generalizes this notion to stacks?

Since we can always pass from a stack to an equivalent sheaf, one option might be to say the stalk of a stack is (any category equivalent to) the stalk of the associated sheaf of categories. This seems a little roundabout, though? Is there a more intrinsic definition of these categories?