The rapid adoption of data-driven methods in biomedicine has intensified concerns over privacy, governance, and regulation, limiting raw data sharing and hindering the assembly of representative cohorts for clinically relevant AI. This landscape necessitates practical, efficient privacy solutions, as cryptographic defenses often impose heavy overhead and differential privacy can degrade performance, leading to sub-optimal outcomes in real-world settings. Here, we present a lightweight federated learning method, INFL, based on Implicit Neural Representations that addresses these challenges. Our approach integrates plug-and-play, coordinate-conditioned modules into client models, embeds a secret key directly into the architecture, and supports seamless aggregation across heterogeneous sites. Across diverse biomedical omics tasks, including cohort-scale classification in bulk proteomics, regression for perturbation prediction in single-cell transcriptomics, and clustering in spatial transcriptomics and multi-omics with both public and private data, we demonstrate that INFL achieves strong, controllable privacy while maintaining utility, preserving the performance necessary for downstream scientific and clinical applications.