Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space.

Q1:Can this tensor product be the trivial vector spacefor non-zero $M$, $N$? If yes, what are examples where this happens?

Q2:Are there criteria on $M$ and $N$ which ensure that the tensor product is non-zero?

*Edit: Thanks for the answers below. It seems that Q2 is too broad to answer in general. However, I am mostly interested in the case of an infinite-dimensional Clifford algebra; in other words, $A$ is central simple.*