This position paper focuses on the problem of building dialogue systems for people who have lost the ability to communicate via speech, e.g., patients of locked-in syndrome or severely disabled people. In order for such people to communicate to other people and computers, dialogue systems that are based on brain responses to (imagined) speech are needed. A speech-based dialogue system typically consists of an automatic speech recognition module and a speech synthesis module. In order to build a dialogue system that is able to work on the basis of brain signals, a system needs to be developed that is able to recognize speech imagined by a person and can synthesize speech from imagined speech. This paper proposes combining new and emerging technology on neural speech recognition and auditory stimulus construction from brain signals to build brain signal-based dialogue systems. Such systems have a potentially large impact on society.