  We prove existence and uniqueness of solutions to a class of stochastic
semilinear evolution equations with a monotone nonlinear drift term and
multiplicative noise, considerably extending corresponding results obtained in
previous work of ours. In particular, we assume the initial datum to be only
measurable and we allow the diffusion coefficient to be locally
Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the
finiteness of the $p$-th moment of solutions depends on the integrability of
the initial datum, in the whole range $p \in ]0,\infty[$. Lipschitz continuity
of the solution map in $p$-th moment is established, under a Lipschitz
continuity assumption on the diffusion coefficient, in the even larger range $p
\in [0,\infty[$. A key role is played by an It\^o formula for the square of the
norm in the variational setting for processes satisfying minimal integrability
conditions, which yields pathwise continuity of solutions. Moreover, we show
how the regularity of the initial datum and of the diffusion coefficient
improves the regularity of the solution and, if applicable, of the invariant
measures.
