We study convergence properties of opinion dynamics with local interactions and limited information exchange. We adopt a general model where the agents update their opinions in rounds to a weighted average of the opinions in their neighborhoods. For fixed neighborhoods, we present a simple randomized protocol that converges in expectation to the stable state of the Friedkin-Johnsen model. For opinion-dependent neighborhoods, we show that the Hegselmann-Krause model converges to a stable state if each agent's neighborhood is restricted either to a subset of her acquaintances or to a small random subset of agents. Our experimental findings indicate that for a wide range of parameters, the convergence time and the number of opinion clusters of the neighborhood-restricted variants are comparable to those of the standard Hegselmann-Krause model.