This work develops an effective distributed algorithm for the solution of stochastic optimization problems that involve partial coupling among both local constraints and local cost functions. While the collection of networked agents is interested in discovering a global model, the individual agents are sensing data that is only dependent on parts of the model. Moreover, different agents may be dependent on different subsets of the model. In this way, cooperation is justified and also necessary to enable recovery of the global information. In view of the local constraints, we show how to relax the optimization problem to a penalized form, and how to enable cooperation among neighboring agents. We establish mean-square-error convergence of the resulting strategy for sufficiently small step-sizes and large penalty factors. We also illustrate performance by means of simulations.