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We present a framework for the design of coding mechanisms that allow remotely operating anomaly detectors in a privacy-preserving manner. We consider the following problem setup. A remote station seeks to identify anomalies based on system input-output signals transmitted over communication networks. However, it is not desired to disclose true data of the system operation as it can be used to infer private information. To prevent adversaries from eavesdropping on the network or at the remote station itself to access private data, we propose a privacy-preserving coding scheme to distort signals before transmission. As a next step, we design a new anomaly detector that runs on distorted signals and produces distorted diagnostics signals, and a decoding scheme that allows extracting true diagnostics data from distorted signals without error. The proposed scheme is built on the synergy of matrix encryption and system Immersion and Invariance (I&I) tools from control theory. The idea is to immerse the anomaly detector into a higher-dimensional system (the so-called target system). The dynamics of the target system is designed such that: the trajectories of the original anomaly detector are immersed/embedded in its trajectories, it works on randomly encoded input-output signals, and produces an encoded version of the original anomaly detector alarm signals, which are decoded to extract the original alarm at the user side. We show that the proposed privacy-preserving scheme provides the same anomaly detection performance as standard Kalman filter-based chi-squared anomaly detectors while revealing no information about system data.