Unmanned Aerial Vehicles (UAVs, or drones) have revolutionized modern media production. Being rapidly deployable “flying cameras”, they can easily capture aesthetically pleasing aerial footage of static or moving filming targets/subjects. Current approaches rely either on manual UAV/gimbal control by human experts or on a combination of complex computer vision algorithms and hardware configurations for automating the flight+flying process. This paper explores an efficient Deep Reinforcement Learning (DRL) alternative, which implicitly merges the target detection and path planning steps into a single algorithm. To achieve this, a baseline DRL approach is augmented with a novel policy distillation component, which transfers knowledge from a suitable, semi-expert Model Predictive Control (MPC) controller into the DRL agent. Thus, the latter is able to autonomously execute a specific UAV cinematography task with purely visual input. Unlike the MPC controller, the proposed DRL agent does not need to know the 3D world position of the filming target during inference. Experiments conducted in a photorealistic simulator showcase superior performance and training speed compared to the baseline agent while surpassing the MPC controller in terms of visual occlusion avoidance.