We prove existence and regularity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on $L^2(G)$, where $G$ is an open bounded domain in $\mathbb{R}^d$ with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, as well as on monotonicity techniques and a maximum principle for stochastic evolution equations. (Partly based on joint work with L. Quer-Sardanyons)