This paper regards the coordination of networked systems, studied in the framework of hybrid dynamical systems. We design a coordination scheme which combines the use of ternary controllers with a self-triggered communication policy. The communication policy requires the agents to measure, at each sampling time, the difference between their states and those of their neighbors. The collected information is then used to update the control and determine the following sampling time. We show that the proposed scheme ensures finite-time convergence to a neighborhood of a consensus state: the coordination scheme does not require the agents to share a global clock, but allows them to rely on local clocks. We then study the robustness of the proposed self-triggered coordination system with respect to skews in the agents' local clocks, to delays, and to limited precision in communication. Furthermore, we present two significant variations of our scheme. First, assuming a global clock to be available, we design a time-varying controller which asymptotically drives the system to consensus. The assumption of a global clock is then discussed, and relaxed to a certain extent. Second, we adapt our framework to a communication model in which each agent polls its neighbors separately, instead of polling all of them simultaneously. This communication policy actually leads to a self-triggered "gossip" coordination system.