Artificial intelligence (AI) agents such as chatbots and personal AI assistants are increasingly popular. These technologies raise new privacy concerns beyond those posed by other AI systems or information technologies. For example, anthropomorphic features of AI chatbots may invite users to disclose more information with these systems than they would otherwise, especially when users interact with chatbots in relationship-like ways. In this paper, we aim to develop a framework for assessing the distinctive privacy ramifications of AI agents, especially as humans begin to interact with them in relationship-like ways. In particular, we draw from prominent theories of privacy and results from human relational psychology to better understand how AI agents may affect human behavior and the flow of personal information. We then assess how these effects could bear on eight distinct values of privacy, such as autonomy, the value of forming and maintaining relationships, security from harm, and more.