We consider state estimation for multiple plants across a shared communication network. Each linear time-invariant plant transmits information through the common network according to either a time-triggered or an event-triggered rule. For an event-triggered algorithm with carrier sense multiple access (CSMA), each plant is assumed to access the network based on a priority mechanism. For a time-triggered algorithm combined with time division multiple access (TDMA), each plant uses the network according to a schedule that is decided a priori. Performance in terms of the communication frequency and the estimation error covariance is analytically characterized for event-triggered schemes. Using these characterizations, we show that event-triggered schemes may perform worse than time-triggered schemes when the effect of the communication network is explicitly considered.