Remote state estimation problems in the presence of an eavesdropper have recently been studied. In this setup, a sensor transmits over a random packet dropping link to a remote estimator, which at the same time can be randomly overheard by an eavesdropper. For i.i.d. packet dropping links to the remote estimator and to the eavesdropper, it has been shown that with unstable systems one can keep the expected estimation error covariance bounded, while the expected eavesdropper error covariance becomes unbounded in the infinite horizon. In this paper we show that the same behaviour can be achieved when transmission of local state estimates occur over a Markovian packet dropping link, and eavesdropping occurs according to another Markovian packet dropping link.