Despite evidence from cognitive sciences that larger groups of speakers tend to develop more structured languages in human communication, scaling up to populations has failed to yield significant benefits in emergent multi-agent communication. In this paper we advocate for an alternate population-level training paradigm for referential games based on the idea of "partitioning" the agents into sender-receiver pairs and limiting co-adaptation across pairs. We show that this results in optimizing a different objective at the population level, where agents maximize (1) their respective "internal" communication accuracy and (2) some measure of alignment between agents. In experiments, we find that this leads to the emergence of languages that are significantly more compositional. Moreover, when agents are trained in populations that are not fully connected (ie. not all agent pairs interact at training time), this approach reduces multi-linguality and improves zero-shot communication with new agents (ie. agents are able to communicate successfully with other agents outside their training partners).