This paper develops a distributed variance-reduced strategy for a collection of interacting agents that are connected by a graph topology. The resulting diffusion-AVRG (where AVRG stands for "amortized variance-reduced gradient") algorithm is shown to have linear convergence to the exact solution, and is more memory efficient than other alternative algorithms. When a batch implementation is employed, it is observed in simulations that diffusion-AVRG is more computationally efficient than exact diffusion or EXTRA, while maintaining almost the same communication efficiency.