This question is related to that one Commutation of tensor products with inverse limits in a specific case where it received a (partial) answer.

I can complement a bit: in any case, $A$ is supposed a commutative ring with unit and $1\not= 0$.

When $A$ is a domain (no zero divisor) and if $A^X\otimes_A A^Y$ is torsion-free (which I do not know in general) then the natural arrow $f\otimes g\mapsto ((x,y)\mapsto f(x)g(y))$ $$ A^X\otimes_A A^Y\rightarrow A^{X\times Y} $$ is an embedding. Who knows more ? (I am specially interested in ness and suff conditions).