Millimeter-wave wireless networks offer high throughput and can (ideally) prevent eavesdropping attacks using narrow, directional beams. Unfortunately, imperfections in physical hardware mean today's antenna arrays all exhibit side lobes, signals that carry the same sensitive data as the main lobe. Our work presents results of the first experimental study of the security properties of mmWave transmissions against side-lobe eavesdropping attacks. We show that these attacks on mmWave links are highly effective in both indoor and outdoor settings, and they cannot be eliminated by improved hardware or currently proposed defenses.