  We show that the complete Segal model structure extends to a model structure
on bimplicial presheaves on a small site $\mathscr{C}$, for which the weak
equivalences are local (or stalkwise) weak equivalences. This model structure
can be realized as a left Bousfield localization of the Jardine model structure
on the simplicial presheaves on a site $\mathscr{C}/ \Delta^{op}$. Furthermore,
it is shown that this model structure is Quillen equivalent to the model
structure of the author's previous preprint entitled 'the Local Joyal Model
Structure'. This Quillen equivalence extends an equivalence between the
complete Segal space and Joyal model structures, due to Joyal and Tierney.
