Say $\mathcal{C}$ is a strict monoidal abelian category and $A$ is a *coalgebra* object in $\mathcal{C}$, with left co-modules $M$ and right co-module $N$ (also in $\mathcal{C}$). Then we have a notion of (derived) tensor product $$M\overset{R}{\otimes} {}^{co}_A N,$$ (note that we are taking a right derived functor because we are in the co-situation of the left derived functor for usual tensor product), which can now be resolved as $$M\otimes_{\mathcal{C}}N\to M\otimes_{\mathcal{C}}A\otimes_{\mathcal{C}}N\to \ldots\to M\otimes_{\mathcal{C}}A\otimes_{\mathcal{C}}\cdots\otimes_{\mathcal{C}}A\otimes_{\mathcal{C}}N\to \ldots$$ with maps given by comultiplication and coaction. (Here I'm assuming all objects involved are flat with respect to taking tensor product in $\mathcal{C}$).

Now say that $\mathcal{C}$ is the monoidal category of $R$ bimodules for some (associative, not necessarily commutative) *algebra* R. Then the expression above reads
$$M\otimes_{R}N\to M\otimes_{R}A\otimes_{R}N\to \ldots\to M\otimes_{R}A\otimes_{R}\cdots\otimes_{R}A\otimes_{R}N\to \ldots.$$
The key point that I want to point out is that this complex uses the entire $R$-bimodule structure on $A$ (and its compatibility with comultiplication), but only half of the $R$-bimodule structure on $M$ and $N$. In particular, such an operation would be defined for any right $A$-comodule $M$ with just **right** $R$-module structure and left $A$-comodule $N$ with just **left** $R$-module structure, so long as the $A$-comodule structures and $R$-module structures satisfy some compatibility. For any such pair $M$, $N$ it should be possible to write down $$M\overset{?}{\otimes} N$$ computed by the same complex.

I'm curious about a formal algebraic setting that captures this data. Note that one thing I could do, given a right $A$-comodule $M$ with compatible right $R$-action and a left $A$-comodule $N$ with compatible left $R$-action would be to form the $A$-co-bimodule $$N\otimes_k M$$ (which would now be an $R$-bimodule) and take its co-Hochschild homology $$M\overset{?}{\otimes} N: = \operatorname{coHH}_*^{R\mathrm{-Bimod}}(A, N\otimes M).$$ However, this doesn't quite satisfy me since it loses the separate nature of $M$ and $N$. It feels like there should be some other algebraic structure at play that is perhaps a generalization of something in the theory of Hopf algebroids. What is it?