We provide a comprehensive solution to the estimation problem of the state for a linear time-invariant system in a distributed fashion over networks that allow only intermittent information transmission. By attaching to each node an observer that employs information received from its neighbors triggered by asynchronous communication events, we propose a distributed state observer that guarantees global exponential stability of the zero estimation error set. The design of parameters is formulated as linear matrix inequalities. A thorough robustness analysis of the proposed observer to unmodeled dynamics, unknown communication times, as well as measurement and communication noise characterized in terms of input-to-state stability is presented. These properties of the proposed observer are shown analytically and validated numerically.