Edit: I know the question is very vague; this is partly due to the fact that I stumbled upon this map rather accidentaly and that I just wondered what can be said about this. So if you have some thoughts about this, please share them :)
Let $R$ be a unital ring, possibly non-commutative. Let $M$ be a right $R$-module and $N$ be a left $R$-module. Then by the basic properties of the tensor product, we know that $$ -\otimes_R-:\operatorname{End}_R(M,M)\times\operatorname{End}_R(N,N)\to \operatorname{End}_{\mathbb{Z}}(M\otimes_R N,M\otimes_R N)\\ (f,g)\mapsto f\otimes_R g $$ is additive in both arguments, and multiplicative in the sense that $(f\circ f')\otimes_R (g\circ g')=(f\otimes_R g)\circ(f'\otimes_R g')$. By the universal property of the tensor product of $\mathbb{Z}$-algebras, i.e. rings, this induces a ring homomorphism $$ \hat{\otimes}_R:\operatorname{End}_R(M,M)\otimes_{\mathbb{z}}\operatorname{End}_R(N,N)\to \operatorname{End}_{\mathbb{Z}}(M\otimes_R N,M\otimes_R N)\\ f\otimes g\mapsto f\otimes_R g. $$ Can anything interesting be said about this ring homomorphism? E.g. is it injective? Or surjective? Or even bijective? And if not, what if we place stronger conditions on $R$? Thanks for sharing any information that comes to your mind.