The ability of large language models (LLMs) to transform, interpret, and comprehend vast quantities of heterogeneous data presents a significant opportunity to enhance data-driven care delivery. However, the sensitive nature of protected health information (PHI) raises valid concerns about data privacy and trust in remote LLM platforms. In addition, the cost associated with cloud-based artificial intelligence (AI) services continues to impede widespread adoption. To address these challenges, we propose a shift in the LLM execution environment from opaque, centralized cloud providers to a decentralized and dynamic fog computing architecture. By executing open-weight LLMs in more trusted environments, such as the user's edge device or a fog layer within a local network, we aim to mitigate the privacy, trust, and financial challenges associated with cloud-based LLMs. We further present SpeziLLM, an open-source framework designed to facilitate rapid and seamless leveraging of different LLM execution layers and lowering barriers to LLM integration in digital health applications. We demonstrate SpeziLLM's broad applicability across six digital health applications, showcasing its versatility in various healthcare settings.