It seems to me that if $X$ is not factorial than one must be careful.
For instance, take the weighted projective space $X=\mathbb{P}(1,1,2)$, and consider the sheaf $\mathcal{O}_X(1)$, whose construction can be found for instance in Dolgachev's paper "Weighted projective varieties". Abstractely, $X$ is isomorphic to a quadric cone in $\mathbb{P}^3$, and the sheaf $\mathcal{O}_X(1)$ has only two sections, so it is not locally free (in fact, it corresponds to a single line in the cone). However, $\mathcal{O}_X(2)$ is locally free, since it corresponds to a hyperplane section (in fact, $X$ is 2-factorial); let $\mathcal{O}_X(-2):=\mathcal{O}_X(2)^*$.
Finally, take
$M:=\mathcal{O}_X(-2) \otimes \mathcal{O}_X(1), \quad N:=\mathcal{O}(1)$.
Then $M \otimes N \cong \mathcal{O}_X$, but $M$ and $N$ are not locally free.