Large language models (LLMs) have shown promising potential for next Point-of-Interest (POI) recommendation. However, existing methods only perform direct zero-shot prompting, leading to ineffective extraction of user preferences, insufficient injection of collaborative signals, and a lack of user privacy protection. As such, we propose a novel Multitask Reflective Large Language Model for Privacy-preserving Next POI Recommendation (MRP-LLM), aiming to exploit LLMs for better next POI recommendation while preserving user privacy. Specifically, the Multitask Reflective Preference Extraction Module first utilizes LLMs to distill each user's fine-grained (i.e., categorical, temporal, and spatial) preferences into a knowledge base (KB). The Neighbor Preference Retrieval Module retrieves and summarizes the preferences of similar users from the KB to obtain collaborative signals. Subsequently, aggregating the user's preferences with those of similar users, the Multitask Next POI Recommendation Module generates the next POI recommendations via multitask prompting. Meanwhile, during data collection, a Privacy Transmission Module is specifically devised to preserve sensitive POI data. Extensive experiments on three real-world datasets demonstrate the efficacy of our proposed MRP-LLM in providing more accurate next POI recommendations with user privacy preserved.