Kickstarting deep reinforcement learning algorithms facilitate a teacher-student relationship among the agents and allow for a well-performing teacher to share demonstrations with a student to expedite the student's training. However, despite the known benefits, the demonstrations may contain sensitive information about the teacher's training data and existing kickstarting methods do not take any measures to protect it. Therefore, we use the framework of differential privacy to develop a mechanism that securely shares the teacher's demonstrations with the student. The mechanism allows for the teacher to decide upon the accuracy of its demonstrations with respect to the privacy budget that it consumes, thereby granting the teacher full control over its data privacy. We then develop a kickstarted deep reinforcement learning algorithm for the student that is privacy-aware because we calibrate its objective with the parameters of the teacher's privacy mechanism. The privacy-aware design of the algorithm makes it possible to kickstart the student's learning despite the perturbations induced by the privacy mechanism. From numerical experiments, we highlight three empirical results: (i) the algorithm succeeds in expediting the student's learning, (ii) the student converges to a performance level that was not possible without the demonstrations, and (iii) the student maintains its enhanced performance even after the teacher stops sharing useful demonstrations due to its privacy budget constraints.