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We present a continuous monitoring system for intermediate-scale quantum processors that allows extracting estimates of noisy native gate and read-out measurements based on the set of executed quantum circuits and resulting measurement outcomes. In contrast to standard approaches for calibration and benchmarking quantum processors, the executed circuits, which are input to the monitoring system, are assumed to be out of any control. We provide the results of applying our system to the synthetically generated data obtained from a quantum emulator, as well as to the experimental data collected from a publicly accessible cloud-based quantum processor. In the both cases, we demonstrate that the developed approach provides valuable results about inherent noises of emulators/processors. Considering that our approach uses only already accessible data from implemented circuits without the need to run additional algorithms, the monitoring system can complement existing approaches. We expect that our monitoring system can become a useful tool for various quantum computers in the near-term horizon, including publicly accessible cloud-based platforms, and reduce resources that are required for their benchmarking and calibration.