Monitoring the respiratory rate is crucial for helping us identify respiratory disorders. Devices for conventional respiratory monitoring are inconvenient and scarcely available. Recent research has demonstrated the ability of non-contact technologies, such as photoplethysmography and infrared thermography, to gather respiratory signals from the face and monitor breathing. However, the current non-contact respiratory monitoring techniques have poor accuracy because they are sensitive to environmental influences like lighting and motion artifacts. Furthermore, frequent contact between users and the cloud in real-world medical application settings might cause service request delays and potentially the loss of personal data. We proposed a non-contact respiratory rate monitoring system with a cooperative three-layer design to increase the precision of respiratory monitoring and decrease data transmission latency. To reduce data transmission and network latency, our three-tier architecture layer-by-layer decomposes the computing tasks of respiration monitoring. Moreover, we improved the accuracy of respiratory monitoring by designing a target tracking algorithm and an algorithm for eliminating false peaks to extract high-quality respiratory signals. By gathering the data and choosing several regions of interest on the face, we were able to extract the respiration signal and investigate how different regions affected the monitoring of respiration. The results of the experiment indicate that when the nasal region is used to extract the respiratory signal, it performs experimentally best. Our approach performs better than rival approaches while transferring fewer data.