Let $R$ be an algebra over complex numbers. Let $N$ be a module over $R$. We can define the algebra $D(N)$ of differential operators $N \rightarrow N$ using Grothendieck’s approach as follows: we define it inductively saying that $D_{-1}(N)=\{0\}$ and $D_n(N) \subset \operatorname{End}(N)$ consist of maps $d\colon N \rightarrow N$ such that $[d,f] \in D_{n-1}(N)$ for every $f \in R$.

Now assume that we are given two $R$-modules $N_1,\, N_2$ and also let $M_1,\, M_2$ be modules over $D(N_1),\, D(N_2)$ respectively.

Question: is it true (in this generality) that the tensor product $M_1 \otimes_R M_2$ is a module over $D(N_1 \otimes_R N_2)$?