Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield localisation of the injective (= Heller) model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ with respect to the Čech hypercovers of $(\mathcal{C}, J)$, or equivalently, the $J$-covering sieves. This is a presentation of the $(\infty, 1)$-category of sheaves of $\infty$-groupoids on $(\mathcal{C}, J)$.
Question. Is there an explicit description of the Čech-local equivalences of (simplicial pre)sheaves on $(\mathcal{C}, J)$, i.e. the weak equivalences in the Čech model structure?
The theory of left Bousfield localisation says that a morphism $f : X \to Y$ is a Čech-local equivalence if and only if the induced morphism $$f^* : \underline{\mathrm{Hom}}(Y, Z) \to \underline{\mathrm{Hom}}(X, Z)$$ is a weak homotopy equivalence of simplicial sets for all Čech-local injective-fibrant simplicial presheaves on $(\mathcal{C}, J)$. In particular, a Čech-local equivalence between Čech-fibrant sheaves on $(\mathcal{C}, J)$ is just a componentwise weak homotopy equivalence.
Although the above is elegant in some sense, I would like to understand what is going on at the level of simplicial presheaves. For contrast, consider the Jardine model structure: there, the weak equivalences are easily described as the morphisms that induce isomorphisms of sheaves of homotopy groups – so this is a notion with eminent geometric meaning. Unfortunately, not every weak equivalence in the Jardine model structure is a Čech-local equivalence. So where is the difference?
