It is independent, provided the two sites of definition, also yield the same infinity topos (so e.g. when they both have finite limits, or the infinity-topos is hypercomplete). Given two simplicial objects $F$ and $G$ of the topos of $\mathcal{E}$, so $$F,G:\Delta^{op} \to \mathcal{E},$$ consider the embedding $$\theta:\mathcal{E} \hookrightarrow Sh_\infty\left(\mathcal{E}\right),$$ where I am writing $Sh_\infty\left(\mathcal{E}\right)$ for the $1$-localic infinity-topos corresponding to $\mathcal{E}.$ I claim that a map $f:F \to G$ is a weak equivalences if and only if the induced map between $\operatorname{colim} \theta \circ F \to \operatorname{colim} \theta \circ G$ is an equivalence in $Sh_\infty\left(\mathcal{E}\right)$.
Proof: $f$ is a weak equivalence if and only if it is is one when considered as a map of simplicial presheaves, which is if and only if it becomes one in the associated infinity-topos. Choose a site $C$. In simplicial presheaves on $C$ with the global model structure, the homotopy colimit of $\theta \circ F$ is $F$ itself, which can be computed as the diagonal of the bisimplicial presheaf which is constant in one direction. Hence, the colimit of $\theta \circ F$ is the infinity sheaf associated to $F$.