  Given Hilbert spaces $H_1,H_2,H_3$, we consider bilinear maps defined on the
cartesian product $S^2(H_2,H_3)\times S^2(H_1,H_2)$ of spaces of
Hilbert-Schmidt operators and valued in either the space $B(H_1,H_3)$ of
bounded operators, or in the space $S^1(H_1,H_3)$ of trace class operators. We
introduce modular properties of such maps with respect to the commutants of von
Neumann algebras $M_i\subset B(H_i)$, $i=1,2,3$, as well as an appropriate
notion of complete boundedness for such maps. We characterize completely
bounded module maps $u\colon S^2(H_2,H_3)\times S^2(H_1,H_2)\to B(H_1,H_3)$ by
the membership of a natural symbol of $u$ to the von Neumann algebra tensor
product $M_1\overline{\otimes} M_2^{op}\overline{\otimes} M_3$. In the case
when $M_2$ is injective, we characterize completely bounded module maps
$u\colon S^2(H_2,H_3)\times S^2(H_1,H_2)\to S^1(H_1,H_3)$ by a weak
factorization property, which extends to the bilinear setting a famous
description of bimodule linear mappings going back to Haagerup,
Effros-Kishimoto, Smith and Blecher-Smith. We make crucial use of a theorem of
Sinclair-Smith on completely bounded bilinear maps valued in an injective von
Neumann algebra, and provide a new proof of it, based on Hilbert $C^*$-modules.
