Whenever I'm doing sheaf things and I have a construction that involves sheafifying, I find it convenient to think "the thing that has the same stalks but sections must be locally trivial sections to espace etale" - it's a nice way to think of/characterize sheafification, sections of sheaves, define pullbacks, etc.
Can we do the same thing on a general, reasonable site (e.g. etale, flat, etc), i.e. for say $\mathscr{F}$ a sheaf on a scheme $X$ on the etale site, can we form something like $\coprod \mathscr{F}_x$ as $x$ runs over geometric points, suitably "Grothendieck topologized" so that the Grothendieck continuous sections over $U \to X$ are precisely $\mathscr{F}(U \to X)$?