Disentangling shared network-level dynamics from region-specific activity is a central challenge in modeling multi-region neural data. We introduce SPIRE (Shared-Private Inter-Regional Encoder), a deep multi-encoder autoencoder that factorizes recordings into shared and private latent subspaces with novel alignment and disentanglement losses. Trained solely on baseline data, SPIRE robustly recovers cross-regional structure and reveals how external perturbations reorganize it. On synthetic benchmarks with ground-truth latents, SPIRE outperforms classical probabilistic models under nonlinear distortions and temporal misalignments. Applied to intracranial deep brain stimulation (DBS) recordings, SPIRE shows that shared latents reliably encode stimulation-specific signatures that generalize across sites and frequencies. These results establish SPIRE as a practical, reproducible tool for analyzing multi-region neural dynamics under stimulation.