Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same topos of sheaves. So the little étale topos is a well-defined category.
Now my questions are:
1) Can we please drop the quasiseparatedness hypothesis in some way? I'm afraid it is necessary to convert "locally of finite type+affine" into "of finite type" in the proof, but...
2) Does a similar argument work for the big étale site (and for the big Zariski site)? The point is that we lose that everything is locally finitely presented, so I think something more should be necessary, but I hope the conclusion is true as well.
Thank you in advance.