You are confusing two completely different notions of "separated".
A presheaf of sets $F$ on a site is called "separated" if whenever $\{U_i \to U\}$ is a covering family, $F(U) \to \prod F(U_i)$ is injective. This is the notion that Skjelnes is talking about. The sheafification of a presheaf of sets is often constructed by iterating twice an endofunctor $L$ on presheaves such that $L$ takes arbitrary presheaves to separated presheaves and separated presheaves to sheaves.
There is also the algebro-geometric notion of separated, which corresponds (for schemes over the complex numbers) to being Hausdorff. This is completely unrelated. In particular the line with doubled tangents along the origin is not separated in the algebro-geometric sense. But it is of course separated as a presheaf of sets, since it is an algebraic space, and in particular a sheaf of sets.