This question is related to that one 
http://mathoverflow.net/questions/200442/commutation-of-tensor-products-with-inverse-limits-in-a-specific-case/200443#200443
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).