We study the stochastic Willmore flow and the stochastic surface diffusion flow for closed or non-closed curves on $\mathbb{R}^2$ in this paper. We equivalently formulate them as a stochastic one-phase Stefan problem (or a stochastic free boundary problem) of the curvature, which is parameterized by the arc-length, and the length of the curves. After rewriting the stochastic Stefan problem as a quasilinear parabolic evolution equation, we apply the theory for quasilinear parabolic stochastic evolution equations developed by Agresti and Veraar in 2022 to get the existence and uniqueness of a local strong solution up to a maximal stopping time that is characterized by a blow-up alternative. When the solutions blow up, the corresponding stochastic curve flows either develop singularities or shrink to a point.