This paper investigates joint accuracy and security in semantics-aware remote monitoring, where source updates intended for a legitimate receiver may also be intercepted by an eavesdropper. We define legitimate reconstruction advantage (LRA) as the joint event that the intended receiver correctly reconstructs the current source state using received updates while the eavesdropper reconstructs it incorrectly. Source evolution changes the relevance of received updates at the two receivers, so their marginal reconstruction accuracies do not, in general, determine the probability of this joint event. Under randomized stationary (RS) transmission and zero-order-hold reconstruction at both receivers, we derive an explicit joint stationary distribution and closed-form expressions for LRA probability and persistence for general finite-state ergodic Markov sources. For two-state and uniform-transition sources, we obtain algebraic expressions that enable low-complexity optimization of RS policies. Analytical and numerical results show that maximizing the probability of legitimate advantage need not maximize its persistence. For two-state sources, rare-transmission analysis reveals that, at a fixed sum of transition probabilities, greater transition asymmetry can increase persistence while reducing LRA probability. Moreover, optimizing marginal accuracy and security metrics can select policies that are suboptimal for LRA. Counterintuitively, more frequent transmission or improved legitimate-channel reliability can increase legitimate reconstruction accuracy while reducing LRA probability. Finally, a geofencing application translates prescribed LRA requirements into spatial constraints on eavesdropper locations.

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