Sensitive information, such as knowledge about an individual's personality, can be can be misused to influence behavior (e.g., via personalized messaging). To assess to what extent an individual's personality can be inferred from user interactions with LLM-based conversational agents (CAs), we analyze and quantify related privacy risks of using CAs. We collected actual ChatGPT logs from N=668 participants, containing 62,090 individual chats, and report statistics about the different types of shared data and use cases. We fine-tuned RoBERTa-base text classification models to infer personality traits from CA interactions. The findings show that these models achieve trait inference with accuracy (ternary classification) better than random in multiple cases. For example, for extraversion, accuracy improves by +44% relative to the baseline on interactions for relationships and personal reflection. This research highlights how interactions with CAs pose privacy risks and provides fine-grained insights into the level of risk associated with different types of interactions.