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Neural recording technologies now enable simultaneous recording of population activity across multiple brain regions, motivating the development of data-driven models of communication between recorded brain regions. Existing models can struggle to disentangle communication from the effects of unrecorded regions and local neural population dynamics. Here, we introduce Multi-Region Latent Factor Analysis via Dynamical Systems (MR-LFADS), a sequential variational autoencoder composed of region-specific recurrent networks. MR-LFADS features structured information bottlenecks, data-constrained communication, and unsupervised inference of unobserved inputs--features that specifically support disentangling of inter-regional communication, inputs from unobserved regions, and local population dynamics. MR-LFADS outperforms existing approaches at identifying communication across dozens of simulations of task-trained multi-region networks. Applied to large-scale electrophysiology, MR-LFADS predicts brain-wide effects of circuit perturbations that were not seen during model fitting. These validations on synthetic and real neural data suggest that MR-LFADS could serve as a powerful tool for uncovering the principles of brain-wide information processing.