When users submit queries to Large Language Models (LLMs), their prompts can often contain sensitive data, forcing a difficult choice: Send the query to a powerful proprietary LLM providers to achieving state-of-the-art performance and risk data exposure, or relying on smaller, local models guarantees data privacy but often results in a degradation of task performance. Prior approaches have relied on static pipelines that use LLM rewriting, which shatters linguistic coherence and indiscriminately removes privacy-sensitive information, including task-critical content. We reformulate this challenge (Privacy-Conscious Delegation) as a sequential decision-making problem and introduce a novel reinforcement learning (RL) framework called Privacy-R1 to solve it. Our framework trains an agent to dynamically route text chunks, learning a policy that optimally balances the trade-off between privacy leakage and task performance. It implicitly distinguishes between replaceable Personally Identifiable Information (PII) (which it shields locally) and task-critical PII (which it strategically sends to the remote model for maximal utility). To validate our approach in complex scenarios, we also introduce a new medical dataset with high PII density. Our framework achieves a new state-of-the-art on the privacy-utility frontier, demonstrating the necessity of learned, adaptive policies for deploying LLMs in sensitive environments. Dataset can be found at: https://github.com/zackhuiiiii/Privacy-R1.