In this work, we consider the problem of a network of agents collectively minimizing a sum of convex functions. The agents in our setting can only access their local objective functions and exchange information with their immediate neighbors. Motivated by applications where computation is imperfect, including, but not limited to, empirical risk minimization (ERM) and online learning, we assume that only noisy estimates of the local gradients are available. To tackle this problem, we adapt a class of Nested Distributed Gradient methods (NEAR-DGD) to the stochastic gradient setting. These methods have minimal storage requirements, are communication aware and perform well in settings where gradient computation is costly, while communication is relatively inexpensive. We investigate the convergence properties of our method under standard assumptions for stochastic gradients, i.e. unbiasedness and bounded variance. Our analysis indicates that our method converges to a neighborhood of the optimal solution with a linear rate for local strongly convex functions and appropriate constant steplengths. We also show that distributed optimization with stochastic gradients achieves a noise reduction effect similar to mini-batching, which scales favorably with network size. Finally, we present numerical results to demonstrate the effectiveness of our method.