The recent success of large language models (LLMs) has been largely driven by vast public datasets. However, the next frontier for LLM development lies beyond public data. Much of the world's most valuable information is private, especially in highly regulated sectors such as healthcare and finance, where data include patient histories or customer communications. Unlocking this data could represent a major leap forward, enabling LLMs with deeper domain expertise and stronger real-world utility. Yet, these data cannot be shared because they are distributed across institutions and constrained by privacy, regulatory, and organizational barriers. Moreover, institutional datasets are typically non-independent and identically distributed (non-IID), differing across sites in population characteristics, data modalities, documentation patterns, and task-specific label distributions. In this paper, we demonstrate a practical approach to unlocking private and distributed institutional data for LLM adaptation through federated collaboration across data silos. Built on the Sherpa.ai Federated Learning platform, our framework enables nodes to jointly fine-tune a shared LLM without exchanging private data. We evaluate this approach through a cross-domain benchmark in healthcare and finance, using four closed-ended question answering and classification datasets: MedQA, MedMCQA, FPB, and FiQA-SA. We compare three parameter-efficient fine-tuning (PEFT) strategies-LoRA, QLoRA, and IA3-across pretrained backbones under non-IID settings reflecting institutional data heterogeneity. Our results show that federated fine-tuning performs close to centralized training and outperforms isolated single-institution learning. From a Green AI perspective, QLoRA and IA3 improve efficiency with limited accuracy degradation, supporting federated PEFT as a viable approach for adapting LLMs where data cannot be shared.