I'm intererested in open surjective geometric morphisms induced by fibrations of sites $S\to T$ a la Moerdijk, but as a warm-up, let's consider the case $S \to \ast$. In the case that $S$ is a poset $P$ as naturally turns up in set theory, then every constant presheaf is separated. My naive extrapolation is that this is at least partly due to the fact $P$ has no initial object, so we don't have any alteration there when applying the functor $(-)^+$ to the constant presheaf.

But is this enough in general? Do we need a condition like asking that the topos $Sh(S)$ is localic?

And then I'd like to know the relative case: given a fibration of sites $S\to T$, when are presheaves on $S$ that are given by pullback along the fibration of sheaves on $T$ separated?