Multi-agent systems outperform single agent in complex collaborative tasks. However, in large-scale scenarios, ensuring timely information exchange during decentralized task execution remains a challenge. This work presents an online decentralized coordination scheme for multi-agent systems under complex local tasks and intermittent communication constraints. Unlike existing strategies that enforce all-time or intermittent connectivity, our approach allows agents to join or leave communication networks at aperiodic intervals, as deemed optimal by their online task execution. This scheme concurrently determines local plans and refines the communication strategy, i.e., where and when to communicate as a team. A decentralized potential game is modeled among agents, for which a Nash equilibrium is generated iteratively through online local search. It guarantees local task completion and intermittent communication constraints. Extensive numerical simulations are conducted against several strong baselines.