Let $R$ be a commutative ring and let $A_1$ and $A_2$ be (not necessarily commutative) $R$-algebras. Under which conditions on $A_1$ and $A_2$ is the following true: For every projective $A_1$-module $P_1$ and every projective $A_2$-module $P_2$ we have that $P_1\otimes_R P_2$ is projective as a $A_1\otimes_R A_2$-module?
For instance, what about the case when $A_1$ and $A_2$ are projective over $R$? Or else, does it help if $Tor_R(A_1,A_2)$ vanishes?
I am also looking for counterexamples for the general case.