The property of synchronization between two identical linear time-invariant (LTI) systems connected through a network with stochastically-driven isolated communication events is studied. More precisely, the goal is to design feedback controllers that, using information obtained over such networks, asymptotically drive the values of their state to synchronization and render the synchronization condition Lyapunov stable. To solve this problem, we propose a dynamic controller with hybrid dynamics, namely, the controller exhibits continuous dynamics between communication events while it has variables that jump at such events. Due to the additional continuous and discrete dynamics inherent in the networked systems and communication structure, we utilize a hybrid systems framework to model the closed-loop system. The problem of synchronization is then recast as a set stabilization problem and, by utilizing recent Lyapunov stability tools for hybrid systems, sufficient conditions for asymptotic stability of the synchronization set are provided for two network topologies: a cascade (unidirectional) network and a feedback (bidirectional) communication network with independent transmission instances. Furthermore, we study the robustness of synchronization by considering a class of perturbations on the transmitted data. Numerical examples are provided.