Let $k$ be a field, let $A,B$ be commutative $k$-algebras, and let $M$ be $A\otimes_k B$-module.
Via the maps $A \to A\otimes_k B$ and $B\to A\otimes_k B$, we may regard $M$ as an $A$-module and as a $B$-module. Furthermore, both of this maps are flat, so that if $M$ is flat over $A\otimes_k B$, then it is also flat over $A$ and flat over $B$.
What about the converse? if $M$ is flat over $A$ and flat over $B$, must $M$ be flat over $A\otimes_k B$?