This paper studies a distributed stochastic optimization problem over random networks with imperfect communications subject to a global constraint, which is the intersection of local constraint sets assigned to agents. The global cost function is the sum of local expectation-valued convex cost functions. By incorporating the augmented Lagrangian technique with the projection method, a stochastic approximatio-based distributed primal-dual algorithm is proposed to solve the problem. Each agent updates its estimate by using the local noisy observations of its gradient function and the imperfect information derived from its neighbors. For the constrained problem, the estimates are first shown to be bounded almost surely (a.s.) and then are proved to converge to the optimal solution set a.s. Furthermore, the asymptotic normality and efficiency of the algorithm are addressed for the unconstrained case. The results demonstrate the influence of random networks, communication noises, and gradient errors on the performance of the algorithm. Finally, numerical simulations are presented to demonstrate the theoretic results.