Corollary

If $Z_1$ or $Z_2$ is a separable algebra over the field K (which is, for example, the case when K is perfect), then the A-module $M_1\otimes M_2$ is semisimple.

The rings $Z_1$ and $Z_2$ are isomorphic to products of fields and are therefore reduced rings. In particular, if K is perfect, they are separable algebras over K (V, §15, No. 5, p. 125, Theorem 3). By Proposition 5 of V, §15, No. 2, p. 120, the tensor product of a separable algebra and a reduced algebra is reduced. So Z is a reduced ring, and the corollary follows from Proposition 5, b).