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?][1]
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?][2] (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][3]:

**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).


  [1]: http://mathoverflow.net/questions/8918/is-an-algebraic-space-group-always-a-scheme
  [2]: http://mathoverflow.net/questions/9043/why-is-this-not-an-algebraic-space
  [3]: http://math.berkeley.edu/~anton/index.php?m1=writings