We consider the problem of analyzing the performance of distributed filters for continuous-time linear stochastic systems under certain information constraints. We associate an undirected and connected graph with the measurements of the system, where the nodes have access to partial measurements in continuous time. Each node executes a locally optimally filter based on the available measurements. In addition, a node communicates its estimate to a neighbor at some randomly drawn discrete time instants, and these activation times of the graph edges are governed by independent Poisson counters. When a node gets some information from its neighbor, it resets its state using a convex combination of the available information. Consequently, each node implements a filtering algorithm in the form of a stochastic hybrid system. We derive bounds on expected value of error covariance for each node, and show that they converge to a common value for each node if the mean sampling rates for communication between nodes are large enough.