Identifying how brain regions work together to process inputs and generate behavior is a central goal in systems neuroscience. Recently, there has been a surge of interest in characterizing communication between brain regions and shared variability in multi-region neural recordings. In this work, we use a multi-region latent variable model (LVM) to identify computations shared-across and independent-to specific brain regions in spiking neural population data from two different animals. Here, we use a multi-region latent variable model (LVM) to identify computations shared-across and independent-to specific brain regions in spiking neural population data from two different animals. Our model is an extension of Gaussian Process Factor Analysis with Poisson observations (PGPFA) that separates per-trial neural activity into shared and independent components via a partitioned linear mapping. This model is distinct from previous approaches in that it is 1) not directional, i.e. there is no regression or delay across regions, 2) it uses an appropriate likelihood for spiking data (Poisson), and 3) it does make strong assumptions about the latent state such as linear dynamics. Rather, it characterizes region-specific and global variability based on latent functions characterized only by their smoothness. We use this multi-region PGPFA model to understand shared and independent latent structure during sensorimotor tasks across two neural datasets: one, lateral intra-parietal area (LIP) and frontal eye field (FEF) electrophysiological recordings during a delayed target visual task in macaques, and the second, silicon probe (Neuropixels) recordings from anterior and posterior striatum during an evidence accumulation task in rats. We find that our model is able to simultaneously uncover both ramping-like structure in per region latents, and rotational dynamics in latents shared across regions. We also find that error signals are more prominently seen in shared structure, whereas stimulus preference is transient in shared structure, but sustained in specific brain regions.