There is an increasing shift towards the use of conversational agents, or chatbots, thanks to their inclusion in consumer hardware (e.g. Alexa, Siri and Google Assistant) and the growing number of essential services moving online. A chatbot allows an organisation to deal with a large volume of user queries with minimal overheads, which in turn allows human operators to deal with more complex issues. In this paper we present our work on maximising responsible, sensitive disclosures to chatbots. The paper focuses on two key studies, the first of which surveyed participants to establish the relative sensitivity of a range of disclosures. From this, we found that participants were equally comfortable making financial disclosures to a chatbot as to a human. The second study looked to support the dynamic personalisation of the chatbot in order to improve the disclosures. This was achieved by exploiting behavioural biometrics (keystroke and mouse dynamics) to identify demographic information about anonymous users. The research highlighted that a fusion approach, combining both keyboard and mouse dynamics, was the most reliable predictor of these biographic characteristics.