We investigate the curvature operator of the second kind on product Riemannian manifolds and obtain some optimal rigidity results. For instance, we prove that the universal cover of an n-dimensional nonflat complete locally reducible Riemannian manifold with (Formula presented)-nonnegative (respectively, (Formula presented)-nonpositive) curvature operator of the second kind must be isometric to (Formula presented) (respectively, (Formula presented)) up to scaling. We also prove analogous optimal rigidity results for (Formula presented) and (Formula presented), n1, n2 ≥ 2, among product Riemannian manifolds, as well as for (Formula presented) and (Formula presented), m1, m2 ≥ 1, among product Kähler manifolds. The approach is pointwise and algebraic. © 2024 The Author, under license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.