The proliferation of conversational artificial intelligence (AI) systems, such as chatbots, has sparked widespread privacy concerns. Previous research suggests that privacy perceptions and practices vary across sociocultural contexts. This study examines public and institutional discourses on conversational AI privacy in the United States and China. Semantic network and discourse analyses of privacy-related discussions on Twitter and Weibo reveal divergent patterns. On Twitter, public discourse emphasizes privacy risks and concerns and advocates for systemic changes, while institutional discourse promotes individualistic approaches to privacy protection. Conversely, on Weibo, public discourse is less focused on privacy risks and more on the positive impacts of AI, aligning closely with institutional narratives. These variations are intertwined with the cultural, political, economic, and regulatory contexts of the two countries. Our study underscores the importance of multi-level analysis in comparative privacy research to provide a holistic view of privacy in various contexts.