Let L1 and L2 be nonnegative self-adjoint operators acting on L2 (X1 ) and L2 (X2 ), respectively, where X1 and X2 are spaces of homogeneous type. Assume that L1 and L2 have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces Hpw,L1,L2 (X1 × X2) associated to L1 and L2 , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A∞(X1 × X2). Our main result is that the spaces Hpw,L1,L2 (X1 × X2) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and g∗λ1,λ2 -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L1 and L2 . Our results are new even in the unweighted product setting.