This work presents and studies a distributed algorithm for solving optimization problems over networks where agents have individual costs to minimize subject to subspace constraints that require the minimizers across the network to lie in a low-dimensional subspace. The algorithm consists of two steps: i) a self-learning step where each agent minimizes its own cost using a stochastic gradient update; ii) and a social-learning step where each agent combines the updated estimates from its neighbors using the entries of a combination matrix that converges in the limit to the projection onto the low-dimensional subspace. We obtain analytical formulas that reveal how the step-size, data statistical properties, gradient noise, and subspace constraints influence the network mean-square-error performance. The results also show that in the small step-size regime, the iterates generated by the distributed algorithm achieve the centralized steady-state MSE performance. We provide simulations to illustrate the theoretical findings.