  We prove that a positive proportion of hypersurfaces in products of
projective spaces over $\mathbb{Q}$ are everywhere locally soluble, for almost
all multidegrees and dimensions, as a generalization of a theorem of Poonen and
Voloch. We also study the specific case of genus $1$ curves in $\mathbb{P}^1
\times \mathbb{P}^1$ defined over $\mathbb{Q}$, represented as bidegree
$(2,2)$-forms, and show that the proportion of everywhere locally soluble such
curves is approximately $87.4\%$. The proportion of these curves in
$\mathbb{P}^1 \times \mathbb{P}^1$ soluble over $\mathbb{Q}_p$ is a rational
function of $p$ for each finite prime $p$. Finally, we include some
experimental data on the Hasse principle for these curves.
