A scheme is *separated* if the diagonal inclusion $X \to X \times X$ is a closed immersion. I what to know if there is a good generalization of `separated' for algebraic stacks?

My usual stack reference, Anton Gerashchenko's stack notes, doesn't seem to provide an answer.

In a previous MO question several related notions came up. The most similar is quasi-separated where you require the diagonal to be quasi-compact. You can check wikipedia for some relevant algebraic geometry terminology. How does this compare to separatedness?

The main obstacle that I can see in defining separated for stacks is that the property of a map of schemes $X \to Y$ being separated does not appear to be local in the target. Since maps between affines are separated, it seems that every map of schemes is *locally* separated. This means that we shouldn't expect the usual trick of replacing an algebraic stack by a scheme which covers it to work very well.