  $V$ is a complete intersection scheme in a multiprojective space if it can be
defined by an ideal $I$ with as many generators as $\textrm{codim}(V)$. We
investigate the multigraded regularity of complete intersections scheme in
$\mathbb{P}^n\times \mathbb{P}^m$. We explicitly compute many values of the
Hilbert functions of $0$-dimensional complete intersections. We show that these
values only depend upon $n,m$, and the bidegrees of the generators of $I$. As a
result, we provide a sharp upper bound for the multigraded regularity of
$0$-dimensional complete intersections.
