We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator $L$ being the generator of a transient Dirichlet form on a finite measure space $(E,\mathcal{B},μ)$ and the initial value in $\mathcal{F}_e^*$, which is the dual space of an extended transient Dirichlet space. $L$ and $\mathcal{F}_e^*$ replace the Laplace operator $Δ$ and $H^{-1}$, respectively, in the classical case. This framework includes stochastic fast diffusion equations, stochastic fractional fast diffusion equations, the Zhang model, and apply to cases with $E$ being a manifold, a fractal or a graph. In addition, our results apply to operators $-f(-L)$, where $f$ is a Bernstein function, e.g. $f(λ)=λ^α$ or $f(λ)=(λ+1)^α-1$, $0<α<1$.