An equilibrium is communication-proof if it is unaffected by new opportunities to communicate and renegotiate. We characterize the set of equilibria of coordination games with pre-play communication in which players have private preferences over the coordinated outcomes. The set of communication-proof equilibria is a small and relatively homogeneous subset of the set of qualitatively diverse Bayesian Nash equilibria. Under a communication-proof equilibrium, players never miscoordinate, play their jointly preferred outcome whenever there is one, and communicate only the ordinal part of their preferences. Moreover, such equilibria are robust to changes in players' beliefs and interim Pareto efficientComment: -