Video anomaly detection (VAD) systems are increasingly deployed in safety critical environments and require a large amount of data for accurate detection. However, such data may contain personally identifiable information (PII), including facial cues and sensitive demographic attributes, creating compliance challenges under the EU General Data Protection Regulation (GDPR). In particular, GDPR requires that personal data be limited to what is strictly necessary for a specified processing purpose. To address this, we introduce Only What's Necessary, a privacy-by-design framework for VAD that explicitly controls the amount and type of visual information exposed to the detection pipeline. The framework combines breadth based and depth based data minimization mechanisms to suppress PII while preserving cues relevant to anomaly detection. We evaluate a range of minimization configurations by feeding the minimized videos to both a VAD model and a privacy inference model. We employ two ranking based methods, along with Pareto analysis, to characterize the resulting trade off between privacy and utility. From the non-dominated frontier, we identify sweet spot operating points that minimize personal data exposure with limited degradation in detection performance. Extensive experiments on publicly available datasets demonstrate the effectiveness of the proposed framework.