We prove that a positive proportion of hypersurfaces in products of projective spaces over Q are everywhere locally soluble, for almost all multidegrees and dimensions, as a generalization of a theorem of Poonen and Voloch [PV04]. We also study the specific case of genus 1 curves in P1 x P1 defined over Q, represented as bidegree (2,2)-forms, and show that the proportion of everywhere locally soluble such curves is approximately 87.4%. As in the case of plane cubics [BCF16], the proportion of these curves in P1 x P1 soluble over Qp is a rational function of p for each finite prime p. Finally, we include some experimental data on the Hasse principle for these curves.