arXiv:2604.17133v1 Announce Type: new Abstract: Continuous glucose monitors (CGMs) used in diabetes care collect rich personal health data that could improve day-to-day self-management. However, current patient platforms only offer static summaries which do not support inquisitive user queries. Large language models (LLMs) could enable free-form inquiries about continuous glucose data, but deploying them over sensitive health records raises privacy and accuracy concerns. In this paper, we present CGM-Agent, a privacy-preserving framework for question answering over personal glucose data. In our design, the LLM serves purely as a reasoning engine that selects analytical functions. All computation occurs locally, and personal health data never leaves the user's device. For evaluation, we construct a benchmark of 4,180 questions combining parameterized question templates with real user queries and ground truth derived from deterministic program execution. Evaluating 6 leading LLMs, we find that top models achieve 94\% value accuracy on synthetic queries and 88\% on ambiguous real-world queries. Errors stem primarily from intent and temporal ambiguity rather than computational failures. Additionally, lightweight models achieve competitive performance in our agent design, suggesting opportunities for low-cost deployment. We release our code and benchmark to support future work on trustworthy health agents.