Cells form spatial patterns by coordinating their gene expressions. How a group of mesoscopic numbers (hundreds-to-thousands) of cells, without pre-defined morphogens and spatial organization, self-organizes spatial patterns remains incompletely understood. Of particular importance are dynamic spatial patterns - such as spiral waves that perpetually move and transmit information over macroscopic length-scales. We developed an open-source, expandable software that can simulate a field of cells communicating with any number of cell-secreted molecules in any manner. With it and a theory developed here, we identified all possible “cellular dialogues” - ways of communicating with two diffusing molecules - and core architectures underlying them that enable diverse, self-organized dynamic spatial patterns that we classified. The patterns form despite widely varying cellular response to the molecules, gene-expression noise, and spatial arrangement and motility of cells. Three-stage, “order-fluctuate-settle” process forms dynamic spatial patterns: cells form long-lived whirlpools of wavelets that, through chaos-like interactions, settle into a dynamic spatial pattern. These results provide a blueprint to help identify missing regulatory links for observed dynamic-pattern formations and in building synthetic tissues.