Virtual Mental Health Assistants (VMHAs) continuously evolve to support the overloaded global healthcare system, which receives approximately 60 million primary care visits and 6 million emergency room visits annually. These systems, developed by clinical psychologists, psychiatrists, and AI researchers, are designed to aid in Cognitive Behavioral Therapy (CBT). The main focus of VMHAs is to provide relevant information to mental health professionals (MHPs) and engage in meaningful conversations to support individuals with mental health conditions. However, certain gaps prevent VMHAs from fully delivering on their promise during active communications. One of the gaps is their inability to explain their decisions to patients and MHPs, making conversations less trustworthy. Additionally, VMHAs can be vulnerable in providing unsafe responses to patient queries, further undermining their reliability. In this review, we assess the current state of VMHAs on the grounds of user-level explainability and safety, a set of desired properties for the broader adoption of VMHAs. This includes the examination of ChatGPT, a conversation agent developed on AI-driven models: GPT3.5 and GPT-4, that has been proposed for use in providing mental health services. By harnessing the collaborative and impactful contributions of AI, natural language processing, and the mental health professionals (MHPs) community, the review identifies opportunities for technological progress in VMHAs to ensure their capabilities include explainable and safe behaviors. It also emphasizes the importance of measures to guarantee that these advancements align with the promise of fostering trustworthy conversations.