Cross-hospital failure in chest X-ray models is often attributed to domain shift, yet most work assumes invariance without measuring it. This paper studies how to measure site leakage directly and how that measurement changes conclusions about transfer methods. We study multi-site self-supervised learning (SSL) and feature-level adversarial site confusion for cross-hospital transfer. We pretrain a ResNet-18 on NIH and CheXpert without pathology labels. We then freeze the encoder and train a linear pneumonia classifier on NIH only, evaluating transfer to RSNA. We quantify site leakage using a post hoc linear probe that predicts acquisition site from frozen backbone features $f$ and projection features $z$. Across 3 random seeds, multi-site SSL improves RSNA AUC from 0.6736 $\pm$ 0.0148 (ImageNet initialization) to 0.7804 $\pm$ 0.0197. Adding adversarial site confusion on $f$ reduces measured leakage but does not reliably improve AUC and increases variance. On $f$, site probe accuracy drops from 0.9890 $\pm$ 0.0021 (SSL-only) to 0.8504 $\pm$ 0.0051 (CanonicalF), where chance is 0.50. On $z$, probe accuracy drops from 0.8912 $\pm$ 0.0092 to 0.7810 $\pm$ 0.0250. These results show that measuring leakage changes how transfer methods should be interpreted: multi-site SSL drives transfer, while adversarial confusion exposes the limits of invariance assumptions.