Large language model (LLM)-based AI delegates are increasingly utilized to act on behalf of users, assisting them with a wide range of tasks through conversational interfaces. Despite their advantages, concerns arise regarding the potential risk of privacy leaks, particularly in scenarios involving social interactions. While existing research has focused on protecting privacy by limiting the access of AI delegates to sensitive user information, many social scenarios require disclosing private details to achieve desired outcomes, necessitating a balance between privacy protection and disclosure. To address this challenge, we conduct a pilot study to investigate user preferences for AI delegates across various social relations and task scenarios, and then propose a novel AI delegate system that enables privacy-conscious self-disclosure. Our user study demonstrates that the proposed AI delegate strategically protects privacy, pioneering its use in diverse and dynamic social interactions.