The advent of personal AI systems promises substantial improvements in our daily lives. Trust in these systems is essential and hinges on consumer confidence in responsible data handling. This study examines how transparency regarding AI systems' data collection and processing affects individual data disclosure, serving as a critical indicator of trust in these technologies. In a randomized field experiment with 151 participants and a state-of-the-art AI system for personal use, we show that information about data collection and processing leads to a heightened willingness to share data. It does not seem to matter whether the information about data collection and processing is presented as a lengthy privacy policy form or a shortened, more accessible form. As long as individuals are informed about the data collection and processing of the AI systems, they are more willing to share even rather sensitive data with those systems.