Disclosing sensitive health conditions offers significant benefits at both individual and societal levels. However, patients often face challenges due to concerns about stigma. The use of social robots and chatbots to support sensitive disclosures is gaining traction, especially with the emergence of LLM models. Yet, numerous technical, ethical, privacy, safety, efficacy, and reporting concerns must be carefully addressed in this context. In this position paper, we focus on the example of HIV status disclosure, examining key opportunities, technical considerations, and risks associated with LLM-backed social robotics.