The integration of robotics and automation into self-driving laboratories (SDLs) can introduce additional safety complexities, in addition to those that already apply to conventional research laboratories. Personal protective equipment (PPE) is an essential requirement for ensuring the safety and well-being of workers in laboratories, self-driving or otherwise. Fires are another important risk factor in chemical laboratories. In SDLs, fires that occur close to mobile robots, which use flammable lithium batteries, could have increased severity. Here, we present Chemist Eye, a distributed safety monitoring system designed to enhance situational awareness in SDLs. The system integrates multiple stations equipped with RGB, depth, and infrared cameras, designed to monitor incidents in SDLs. Chemist Eye is also designed to spot workers who have suffered a potential accident or medical emergency, PPE compliance and fire hazards. To do this, Chemist Eye uses decision-making driven by a vision-language model (VLM). Chemist Eye is designed for seamless integration, enabling real-time communication with robots. Based on the VLM recommendations, the system attempts to drive mobile robots away from potential fire locations, exits, or individuals not wearing PPE, and issues audible warnings where necessary. It also integrates with third-party messaging platforms to provide instant notifications to lab personnel. We tested Chemist Eye with real-world data from an SDL equipped with three mobile robots and found that the spotting of possible safety hazards and decision-making performances reached 97 % and 95 %, respectively.