Multi-center research has strengthened the generalizability and reproducibility of scientific findings, yet multi-site inference in modern distributed research networks remains constrained by privacy regulations, heterogeneous site characteristics, and the logistical burden of human-in-the-loop, multi-round communication. In this work, we distill these operational constraints into four mathematical requirements for practical distributed inference and introduce MOSAiC, a unified meta-algorithmic framework that addresses them simultaneously. MOSAiC leverages tensor-train tools from scientific computing to reframe distributed learning as a problem of multivariate function approximation, enabling one-shot communication, robustness to small or irregular sites, accuracy comparable to pooled analysis, and analytic submodel flexibility without additional queries. We demonstrate the feasibility and scalability of the approach through deployments across four institutions and applications in drug relabeling, drug repurposing, and post-market safety surveillance. Beyond these results, MOSAiC offers an operationally grounded foundation for designing next generation federated learning methods – one that aligns statistical rigor with the operational rhythms of real-world scientific collaboration and opens new directions for methodological innovation.