Wearable, implantable, and environmental (WIE) "edge" sensors, being deployed for a broad range of applications including smart healthcare and agriculture, are typically powered by various active (e.g., triboelectric generator, solar cell, etc.) and passive (e.g., drone-supported RFID) energy harvesters. Unfortunately, these active and passive harvesters rely on intermittent sources of energy, and additional circuitry needed to integrate the harvester with the sensor increases design cost and complexity. Interestingly, this problem can be resolved if the sensor can power itself by time-multiplexed harvesting of the energy contained in the signal/noise. These "dual-function" sensors have been previously used in self-powered microphones, cameras, biofuel cells, etc. Since a potentiometric ion-selective electrode (ISE) measures analyte concentration in the form of a voltage signal, in principle, the voltage can be used to self-power the ISE system, including the signal readout and processing circuitry. The principle has already been demonstrated for the self-powered reading of the voltage signal; however, a long-term analysis of the viability and reliability of ISE sensing and harvesting is needed for integration into continuous monitoring edge devices. In this paper, we develop an equivalent circuit model of ISE from electrochemical impedance analysis to quantify its key reliability issues such as voltage stability, sensitivity, and power harvesting capacity by repetitive charging and discharging of a capacitor. The circuit model is verified by a systematic set of laboratory experiments. We show that an unoptimized pH sensor can harvest 0.12 nW/cm(2) while simultaneously measuring the analyte activity for days; the experimental evidence indicates that the sensing and harvesting can continue indefinitely. Using the circuit model as an optimization tool, we experimentally show that power yield can be increased approximately 500 times to 50 nW/cm(2) by replacing the pH sensor with bare Pt wire, demonstrating the potential of such harvesters.