Remote state estimation problems in the presence of eavesdroppers have recently been investigated in the literature. For unstable systems, it has been shown that it is possible to keep the expected estimation error covariance bounded, while the expected eavesdropper error covariance becomes unbounded in the infinite horizon. In this note, we consider an alternative notion of security based on the amount of information revealed to the eavesdropper. Upper and lower bounds on the information revealed are derived. In particular, in the infinite horizon, it is shown that with unstable systems, any transmission policy (within the class of stationary deterministic policies where the sensor at each time step can either transmit its local state estimate or not) which keeps the expected estimation error covariance bounded must always reveal a non-zero expected amount of information to the eavesdropper.