We establish for the first time the explicit curvature formulas for the horizontal and vertical edges of the strong product of two regular graphs. We complement this result with showing that there does not exist an analogous formula for the curvatures of diagonal edges except for a special case, and providing a sharp lower bound for them in terms of the curvatures of the factors. This gives the curvature formulas for all the edges of the product of a complete graph and a regular graph. We also present an accessible and simpler proof of the curvature formulas for all the edges of the Cartesian product of two regular graphs, originally established by Lin, Lu, and Yau [2011].