Projections of bipartite or two-mode networks capture co-occurrences, and are used in diverse fields (e.g., ecology, economics, bibliometrics, politics) to represent unipartite networks that would otherwise be difficult or impossible to measure directly. A key challenge in analyzing such networks is determining whether an observed number of co-occurrences is significant. Several models now exist for doing so and thus for extracting the backbone of bipartite projections, but they have not been directly compared to each other. In this paper, we compare five such models -- fixed fill model (FFM) fixed row model (FRM), fixed column model (FCM), fixed degree sequence model (FDSM), and stochastic degree sequence model (SDSM) -- in terms of accuracy, speed, statistical power, similarity, and community detection. We find that the computationally-fast SDSM offers a statistically conservative but close approximation of the computationally-impractical FDSM under a wide range of conditions, and that it correctly recovers a known community structure even when the signal is weak. Therefore, although each backbone model may have particular applications, we recommend SDSM for extracting the backbone of most bipartite projections.