Let $G$ be a group, and for a $G$-module $A$, denote by $A_G$ the group of coinvariants (i.e., $A_G = \mathbb{Z} \otimes_G A$). In Atiyah's group cohomology text, the following claim is made: for $G$-modules $A,B$, we have $$(A \otimes_{\mathbb{Z}} B)_G \simeq A \otimes_G B,$$ where I the identification is, I presume, one of $G$-modules. But this appears to be false: the action of $G$ on the LHS (the coinvariants) is trivial, whereas the action on the RHS is nontrivial.
Am I correct to say that the above is just a typo, and that the authors probably meant to write $$(A \otimes_{\mathbb{Z}} B)_G \simeq A_G \otimes_{\mathbb{Z}} B_G?$$
No. The identification is as abelian groups. $A \otimes_G B$ does not mean the tensor product you might be familiar with from representation theory; it means the LHS. The second statement you write is false.