Multicellular organisms rely on spatial signaling among cells to drive their organization, development, and response to stimuli. Several models have been proposed to capture the behavior of spatial signaling in multicellular systems, but existing approaches fail to capture both the autonomous behavior of single cells and the interactions of a cell with its neighbors simultaneously. We propose a spatiotemporal model of dynamic cell signaling based on Hawkes processes-self-exciting point processes-that model the signaling processes within a cell and spatial couplings between cells. With this cellular point process (CPP), we capture both the single-cell pathway activation rate and the magnitude and duration of signaling between cells relative to their spatial location. Furthermore, our model captures tissues composed of heterogeneous cell types with different bursting rates and signaling behaviors across multiple signaling proteins. We apply our model to epithelial cell systems that exhibit a range of autonomous and spatial signaling behaviors basally and under pharmacological exposure. Our model identifies known drug-induced signaling deficits, characterizes signaling changes across a wound front, and generalizes to multichannel observations.