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A key property of linguistic conventions is that they hold over an entire community of speakers, allowing us to communicate efficiently even with people we have never met before. At the same time, much of our language use is partner-specific: we know that words may be understood differently by different people based on our shared history. This poses a challenge for accounts of convention formation. Exactly how do agents make the inferential leap to community-wide expectations while maintaining partner-specific knowledge? We propose a hierarchical Bayesian model to explain how speakers and listeners solve this inductive problem. To evaluate our model's predictions, we conducted an experiment where participants played an extended natural-language communication game with different partners in a small community. We examine several measures of generalization and find key signatures of both partner-specificity and community convergence that distinguish our model from alternatives. These results suggest that partner-specificity is not only compatible with the formation of community-wide conventions, but may facilitate it when coupled with a powerful inductive mechanism.