We consider networks of agents cooperating to minimize a global objective, modeled as the aggregate sum of regularized costs that are not required to be differentiable. Since the subgradients of the individual costs cannot generally be assumed to be uniformly bounded, general distributed subgradient techniques are not applicable to these problems. We isolate the requirement of bounded subgradients into the regularizer and use splitting techniques to develop a stochastic proximal diffusion strategy for solving the optimization problem by continuously learning from streaming data. We represent the implementation as the cascade of three operators and invoke Banach's fixed-point theorem to establish that, despite gradient noise, the stochastic implementation is able to converge in the mean-square-error sense within O(μ) from the optimal solution, for a sufficiently small step-size parameter, μ.