Advancements in deep learning have improved anomaly detection in surveillance videos, yet they raise urgent privacy concerns due to the collection of sensitive human data. In this paper, we present a comprehensive analysis of anomaly detection performance under four human anonymization techniques, including blurring, masking, encryption, and avatar replacement, applied to the UCF-Crime dataset. We evaluate four anomaly detection methods, MGFN, UR-DMU, BN-WVAD, and PEL4VAD, on the anonymized UCF-Crime to reveal how each method responds to different obfuscation techniques. Experimental results demonstrate that anomaly detection remains viable under anonymized data and is dependent on the algorithmic design and the learning strategy. For instance, under certain anonymization patterns, such as encryption and masking, some models inadvertently achieve higher AUC performance compared to raw data, due to the strong responsiveness of their algorithmic components to these noise patterns. These results highlight the algorithm-specific sensitivities to anonymization and emphasize the trade-off between preserving privacy and maintaining detection utility. Furthermore, we compare these conventional anonymization techniques with the emerging privacy-by-design solutions, highlighting an often overlooked trade-off between robust privacy protection and utility flexibility. Through comprehensive experiments and analyses, this study provides a compelling benchmark and insights into balancing human privacy with the demands of anomaly detection.