  Let $\gamma_1,\gamma_2$ be a pair of constant-degree irreducible algebraic
curves in $\mathbb{R}^d$. Assume that $\gamma_i$ is neither contained in a
hyperplane nor in a quadric surface in $\mathbb{R}^d$, for each $i=1,2$. We
show that for every pair of $n$-point sets $P_1\subset\gamma_1$ and
$P_2\subset\gamma_2$, the number of distinct distances spanned by $P_1\times
P_2$ is $\Omega(n^{3/2})$, with a constant of proportionality that depends on
${\rm deg}\gamma_1$, ${\rm deg}\gamma_2$, and $d$. This extends earlier results
of Charalambides [Char], Pach and De Zeeuw [PdZ], and Raz [Ra] to the bipartite
version. For the proof we use rigidity theory, and in particular the
description of Bolker and Roth [BR80] for realizations in $\mathbb{R}^d$ of the
complete bipartite graph $K_{m,n}$ that are not infinitesimally rigid.
