In Deligne and Mumford's famous 1969 paper, [The irreducibility of the space of curves of given genus](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1969__36_/PMIHES_1969__36__75_0/PMIHES_1969__36__75_0.pdf), definition 4.6 (that of algebraic stacks) has the following footnote:



> 
> This definition is the "right" one only for quasi-separated stacks. It will however be 
>  sufficient for our purposes.
> 
> 
> 


Note that their definition of an algebraic stack is:


1. A stack on $Sch$ with the etale topology such that
2. The diagonal is representable, and
3. there is a representable etale surjection from a scheme.


and it immediately follows the definition of what it means for a stack to be quasi-separated, but they clearly do not mention quasi-separability here.


What do they mean by the footnote? What is the "right" definition for stacks which are not quasi-separated?



Query on comment in Deligne-Mumford (1969)
I think that what they might have in mind is that for non-quasi-separated Deligne-Mumford algebraic stacks one should not assume that the diagonal is represented by schemes, but by algebraic spaces. For quasi-separated Deligne-Mumford stacks this implies representability by schemes, but this is not true in general. 


At least, this was my guess when I read Deligne-Mumford for the first time, as a graduate student. The first articles by Artin on algebraic spaces appeared in 1969 (a momentous year in algebraic geometry), and Deligne and Mumford must have been aware of his ideas.


