Protein phosphorylation signaling networks play a central role in how cells sense and respond to their environment. Here, we describe the engineering of artificial phosphorylation networks in which "push-pull" motifs-reversible enzymatic phosphorylation cycles consisting of opposing kinase and phosphatase activities-are assembled from modular protein domain parts and then wired together to create synthetic phosphorylation circuits in human cells. We demonstrate that the composability of our design scheme enables model-guided tuning of circuit function and the ability to make diverse network connections; synthetic phosphorylation circuits can be coupled to upstream cell surface receptors to enable fast-timescale sensing of extracellular ligands, while downstream connections can regulate gene expression. We leverage these capabilities to engineer cell-based cytokine controllers that dynamically sense and suppress activated T cells. Our work introduces a generalizable approach for designing and building phosphorylation signaling circuits that enable user-defined sense-and-respond function for diverse biosensing and therapeutic applications.