Through advances in their conversational abilities, chatbots have started to request and process an increasing variety of sensitive personal information. The accurate disclosure of sensitive information is essential where it is used to provide advice and support to users in the healthcare and finance sectors. In this study, we explore users' concerns regarding factors associated with the use of sensitive data by chatbot providers. We surveyed a representative sample of 491 British citizens. Our results show that the user concerns focus on deleting personal information and concerns about their data's inappropriate use. We also identified that individuals were concerned about losing control over their data after a conversation with conversational agents. We found no effect from a user's gender or education but did find an effect from the user's age, with those over 45 being more concerned than those under 45. We also considered the factors that engender trust in a chatbot. Our respondents' primary focus was on the chatbot's technical elements, with factors such as the response quality being identified as the most critical factor. We again found no effect from the user's gender or education level; however, when we considered some social factors (e.g. avatars or perceived 'friendliness'), we found those under 45 years old rated these as more important than those over 45. The paper concludes with a discussion of these results within the context of designing inclusive, digital systems that support a wide range of users.