Many networks in political and social research are bipartite, with edges connecting exclusively across two distinct types of nodes. A common example includes cosponsorship networks, in which legislators are connected indirectly through the bills they support. Yet most existing network models are designed for unipartite networks, where edges can arise between any pair of nodes. However, using a unipartite network model to analyze bipartite networks, as often done in practice, can result in aggregation bias and artificially high-clustering -- a particularly insidious problem when studying the role groups play in network formation. To address these methodological problems, we develop a statistical model of bipartite networks theorized to be generated through group interactions by extending the popular mixed-membership stochastic blockmodel. Our model allows researchers to identify the groups of nodes, within each node type in the bipartite structure, that share common patterns of edge formation. The model also incorporates both node and dyad-level covariates as the predictors of group membership and of observed dyadic relations. We develop an efficient computational algorithm for fitting the model, and apply it to cosponsorship data from the United States Senate. We show that legislators in a Senate that was perfectly split along party lines were able to remain productive and pass major legislation by forming non-partisan, power-brokering coalitions that found common ground through their collaboration on low-stakes bills. We also find evidence for norms of reciprocity, and uncover the substantial role played by policy expertise in the formation of cosponsorships between senators and legislation. We make an open-source software package available that makes it possible for other researchers to uncover similar insights from bipartite networks.