This issue or question came up indirectly in a couple previous posts, which I think you might like to look at. There is indeed a notion of algebraic space which is more general and doesn't require quasi-separatedness (see below). The first such question was Anton's post: Is an Algebraic Space Group Always a Scheme? In that post he asked whether a group object in algebraic spaces is necessarily a scheme. It turns out that the answer depends very heavily on whether the definition of algebraic space requires quasi-sep. or not. If it requires it, then the answer is yes. If not then there are counter examples, which I learned by asking this question Why is This Not an Algebraic Space? (the object in question is a group object in non-quasi-separated algebraic spaces, which is not a scheme).

When I learned the definition of algebraic space (which was some time ago in Martin Olsson class on Stacks at UC Berkeley) it didn't include Quasi-Sep. Here is the definition we used, which I looked up in Anton's wonderful collection of notes:

Definition: An algebraic space over S is a functor $X : (Sch/S)^{op} \to Set$ such that

1. X is a sheaf on the big etale topology on S,
2. $\Delta : X → X \times_S X$ is representable, and
3. there exists an S-scheme $U \to S$ and a surjective etale morphism $U \to X$ (surjective as a map of sheaves).

This issue or question came up indirectly in a couple previous posts, which I think you might like to look at. There is indeed a notion of algebraic space which is more general and doesn't require quasi-separatedness (see below). The first such question was Anton's post: Is an Algebraic Space Group Always a Scheme? In that post he asked whether a group object in algebraic spaces is necessarily a scheme. It turns out that the answer depends very heavily on whether the definition of algebraic space requires quasi-sep. or not. If it requires it, then the answer is yes. If not then there are counter examples, which I learned by asking this question Why is This Not an Algebraic Space? (the object in question is a group object in non-quasi-separated algebraic spaces, which is not a scheme).

When I learned the definition of algebraic space (which was some time ago in Martin Olsson class on Stacks at UC Berkeley) it didn't include Quasi-Sep. Here is the definition we used, which I looked up in Anton's wonderful collection of notes:

Definition: An algebraic space over S is a functor $X : (Sch/S)^{op} \to Set$ such that

1. X is a sheaf on the big etale topology on S,
2. $\Delta : X → X \times_S X$ is representable, and
3. there exists an S-scheme $U \to S$ and a surjective etale morphism $U \to X$ (surjective as a map of sheaves).