Video anomaly detection (VAD) without human monitoring is a complex computer vision task that can have a positive impact on society if implemented successfully. While recent advances have made significant progress in solving this task, most existing approaches overlook a critical real-world concern: privacy. With the increasing popularity of artificial intelligence technologies, it becomes crucial to implement proper AI ethics into their development. Privacy leakage in VAD allows models to pick up and amplify unnecessary biases related to people's personal information, which may lead to undesirable decision making. In this paper, we propose TeD-SPAD, a privacy-aware video anomaly detection framework that destroys visual private information in a self-supervised manner. In particular, we propose the use of a temporally-distinct triplet loss to promote temporally discriminative features, which complements current weakly-supervised VAD methods. Using TeD-SPAD, we achieve a positive trade-off between privacy protection and utility anomaly detection performance on three popular weakly supervised VAD datasets: UCF-Crime, XD-Violence, and ShanghaiTech. Our proposed anonymization model reduces private attribute prediction by 32.25% while only reducing frame-level ROC AUC on the UCF-Crime anomaly detection dataset by 3.69%. Project Page: https://joefioresi718.github.io/TeD-SPAD_webpage/