In this paper, we propose and validate a Joint-Initiative Supervised Autonomy (JISA) framework for Human-Robot Interaction (HRI), in which a robot maintains a measure of its self-confidence (SC) while performing a task, and only prompts the human supervisor for help when its SC drops. At the same time, during task execution, a human supervisor can intervene in the task being performed, based on his/her Situation Awareness (SA). To evaluate the applicability and utility of JISA, it is implemented on two different HRI tasks: grid-based collaborative simultaneous localization and mapping (SLAM) and automated jigsaw puzzle reconstruction. Augmented Reality (AR) (for SLAM) and two-dimensional graphical user interfaces (GUI) (for puzzle reconstruction) are custom-designed to enhance human SA and allow intuitive interaction between the human and the agent. The superiority of the JISA framework is demonstrated in experiments. In SLAM, the superior maps produced by JISA preclude the need for post processing of any SLAM stock maps; furthermore, JISA reduces the required mapping time by approximately 50 percent versus traditional approaches. In automated puzzle reconstruction, the JISA framework outperforms both fully autonomous solutions, as well as those resulting from on-demand human intervention prompted by the agent.