The rapid growth of healthcare data across hospitals, imaging centers, and wearable devices creates opportunities for data-driven clinical decision support, yet strict privacy regulations prevent centralized aggregation of sensitive records. Federated Learning (FL) enables decentralized model training without sharing raw data; however, traditional FL is still shown to degrade performance under heterogeneous client distributions and shows limited adaptability to local environments. This feasibility study presents a Hybrid Federated Learning Framework that combines global model aggregation with local, client-based personalization to promote scalability and accuracy at the local level in multi-institutional contexts. A ResNet-18 backbone was trained on the NIH ChestX-ray14 dataset with patient-level data partitioning. The hybrid FL model achieved mean AUROC scores between .74 and .87, which closely approximated a centrally developed model. Localized, personalized improvements were also found to augment holdout AUROC by 2-6% as compared to a global-only baseline. To bolster privacy, we employed differential privacy on the client-level and showed that moderate differential privacy budgets (ε approximated 2-5) provided a similar level of accuracy as the global model, with less than minimal utility loss. The results suggest that personalized hybrid FL is a secure, privacy-preserving, and scalable framework for healthcare analytics and achieves near-centralized computational performance while maintaining the privacy of patient information. Future work will expand this framework for real-time Internet of Things (IoT) medical devices and utilize communication-efficient aggregation methods and compression techniques, including blockchain-assisted secure protocols.