With the advent of Industry 5.0, manufacturers are increasingly prioritizing worker well-being alongside mass customization. Stress-aware Human-Robot Collaboration (HRC) plays a crucial role in this paradigm, where robots must adapt their behavior to human mental states to improve collaboration fluency and safety. This paper presents a novel framework that integrates Federated Learning (FL) to enable personalized mental state evaluation while preserving user privacy. By leveraging physiological signals, including EEG, ECG, EDA, EMG, and respiration, a multimodal model predicts an operator's stress level, facilitating real-time robot adaptation. The FL-based approach allows distributed on-device training, ensuring data confidentiality while improving model generalization and individual customization. Results demonstrate that the deployment of an FL approach results in a global model with performance in stress prediction accuracy comparable to a centralized training approach. Moreover, FL allows for enhancing personalization, thereby optimizing human-robot interaction in industrial settings, while preserving data privacy. The proposed framework advances privacy-preserving, adaptive robotics to enhance workforce well-being in smart manufacturing.