This position paper presents AVEC (Adaptive Verifiable Edge Control), a framework for bootstrapping privacy for local language models by enforcing privacy at the edge with explicit verifiability for delegated queries. AVEC introduces an adaptive budgeting algorithm that allocates per-query differential privacy parameters based on sensitivity, local confidence, and historical usage, and uses verifiable transformation with on-device integrity checks. We formalize guarantees using Rényi differential privacy with odometer-based accounting, and establish utility ceilings, delegation-leakage bounds, and impossibility results for deterministic gating and hash-only certification. Our evaluation is simulation-based by design to study mechanism behavior and accounting; we do not claim deployment readiness or task-level utility with live LLMs. The contribution is a conceptual architecture and theoretical foundation that chart a pathway for empirical follow-up on privately bootstrapping local LLMs.