As we move toward a future where robots are increasingly part of daily life, the privacy risks associated with interactions, particularly those relying on cloud-based large language models (LLMs), are becoming more pressing. Users may unknowingly share sensitive information in environments, such as homes or hospitals. To explore these risks, we conducted a study with 39 native English speakers using a Furhat robot with an integrated LLM. Participants discussed two moral dilemmas: (i) dishonesty, sharing personal stories of justified lying, and (ii) robot disobedience, discussing whether robots should disobey commands. On average, participants disclosed personal stories 45% of the time when asked in both scenarios. The main reason for non-disclosure was difficulty recalling examples quickly (33.3-56%), rather than reluctance to share (7.2-16%). However, most participants reported a lack of discomfort and concern about sharing personal information with the robot, indicating limited awareness of the privacy risks involved in such disclosures.