Cognition arises from the coordinated interaction of multiple circuits with individual computational roles. While our ability to extract the dynamics underlying circuit computation from population activity recorded in individual areas, understanding how multiple areas jointly support distributed computation remains a challenge. Here we propose a novel multi-region statistical model which reflects a process level description of distributed neural computation. Specifically, the model includes local within-region nonlinear dynamics, potentially driven by task-specific inputs, and communication channels between regions, parameterized through their impulse response. The resulting Multi-Region Dynamical System with Impulse Response communication channels (MRDS-IR) can be fit to data via specialized end-to-end variational inference and learning, designed to exploit the sparsity structure inherent in the model specification. We demonstrate that our approach can recover the true structure of both local computations within areas and information flow between them in complex simulated scenarios reflecting relevant brain computations. Fitting the model to simultaneous population recordings from areas V1 and V2 reveals the structure of the visual stimuli being encoded while recapitulating known properties of feedforward and feedback interactions between visual areas. Overall, our results argue for MRDS-IR as a useful tool for data driven understanding of distributed neural computation.