Cells form spatial patterns by coordinating their gene expressions. How a group of mesoscopic numbers (hundreds to thousands) of cells, without pre-existing morphogen gradients and spatial organization, self-organizes spatial patterns remains poorly understood. Of particular importance are dynamic spatial patterns such as spiral waves that perpetually move and transmit information. We developed an open-source software for simulating a field of cells that communicate by secreting any number of molecules. With this software and a theory, we identified all possible "cellular dialogues"-ways of communicating with two diffusing molecules-that yield diverse dynamic spatial patterns. These patterns emerge despite widely varying responses of cells to the molecules, gene-expression noise, spatial arrangements, and cell movements. A three-stage, "order-fluctuate-settle" process forms dynamic spatial patterns: cells form long-lived whirlpools of wavelets that, following erratic dynamics, settle into a dynamic spatial pattern. Our work helps in identifying gene-regulatory networks that underlie dynamic pattern formations.