Understanding how multiple brain regions interact to produce behavior is a major challenge in systems neuroscience, with many regions causally implicated in common tasks such as sensory processing and decision making. A precise description of interactions between regions remains an open problem. Moreover, neural dynamics are nonlinear and non-stationary. Here, we propose MR-SDS, a multiregion, switching nonlinear state space model that decomposes global dynamics into local and cross-communication components in the latent space. MR-SDS includes directed interactions between brain regions, allowing for estimation of state-dependent communication signals, and accounts for sensory inputs effects, history effects, and heterogeneity across days and animals. We show that our model accurately recovers latent trajectories, vector fields underlying switching nonlinear dynamics, and cross-region communication profiles in three simulations. We then apply our method to two large-scale, multi-region neural datasets involving mouse decision making. The first includes hundreds of neurons per region, recorded simultaneously at single-cell-resolution across 3 distant cortical regions. The second is a mesoscale widefield dataset of 8 adjacent cortical regions imaged across both hemispheres. On these multi-region datasets, our model outperforms existing piece-wise linear multi-region models and reveals multiple distinct dynamical states and a rich set of cross-region communication profiles.