Non-contact technology for monitoring the vital signs of multiple individuals, such as respiration and heartbeat, has been investigated in recent years due to the rising cardiopulmonary morbidity, the risk of disease transmission, and the heavy burden on medical staff. Frequency-modulated continuous wave (FMCW) radars have shown great promise in meeting these needs, even using a single-input-single-output (SISO) setup. However, contemporary techniques for non-contact vital signs monitoring (NCVSM) via SISO FMCW radar, are based on simplistic models and present difficulties in coping with noisy environments containing multiple objects. In this work, we first develop an extended model for multi-person NCVSM via SISO FMCW radar. Then, by utilizing the sparse nature of the modeled signals in conjunction with human-typical cardiopulmonary features, we present accurate localization and NCVSM of multiple individuals in a cluttered scenario, even with only a single channel. Specifically, we provide a joint-sparse recovery mechanism to localize people and develop a robust method for NCVSM called Vital Signs-based Dictionary Recovery (VSDR), which uses a dictionary-based approach to search for the rates of respiration and heartbeat over high-resolution grids corresponding to human cardiopulmonary activity. The advantages of our method are illustrated through examples that combine the proposed model with in-vivo data of 30 individuals. We demonstrate accurate human localization in a noisy scenario that includes both static and vibrating objects and show that our VSDR approach outperforms existing NCVSM techniques based on several statistical metrics. The findings support the widespread use of FMCW radars with the proposed algorithms in healthcare.