The emergence of large language models (LLMs), and their increased use in user-facing systems, has led to substantial privacy concerns. To date, research on these privacy concerns has been model-centered: exploring how LLMs lead to privacy risks like memorization, or can be used to infer personal characteristics about people from their content. We argue that there is a need for more research focusing on the human aspect of these privacy issues: e.g., research on how design paradigms for LLMs affect users' disclosure behaviors, users' mental models and preferences for privacy controls, and the design of tools, systems, and artifacts that empower end-users to reclaim ownership over their personal data. To build usable, efficient, and privacy-friendly systems powered by these models with imperfect privacy properties, our goal is to initiate discussions to outline an agenda for conducting human-centered research on privacy issues in LLM-powered systems. This Special Interest Group (SIG) aims to bring together researchers with backgrounds in usable security and privacy, human-AI collaboration, NLP, or any other related domains to share their perspectives and experiences on this problem, to help our community establish a collective understanding of the challenges, research opportunities, research methods, and strategies to collaborate with researchers outside of HCI.