A simplicial sheaf $A$ is equivalently
in a category of sheaves $Sh(C)$ for some site $C$;
that satisfies degreewise the sheaf condition;
which, when regarded under the equivalence
is degreewise a sheaf.
The Jardine-local model structure on simplicial presheaves restricts to the standard model structure on simplicial sheaves. This restriction is a Quillen equivalence, so that equipped with this model structure $SSh(C)$ is a model for the hypercomplete (infinity,1)-topos over the site $C$.
A discussion of the homotopy theory of simplicial objects in toposes using Cisinski model structures is in
The last part of
is announced to be about simplicial objects in toposes, but that part does not exist yet.
For more see at simplicial presheaf and model structure on simplicial presheaves.