In the early. days of radiology, little was known about the long term effects of radiation, and many xray workers and doctors ended up with cancers. Nowadays, exposure is minimized by safety equipment and improved technology. Radiologist xray exposure is monitored on a regular basis with badges and other detectors.
No! They might, if they didn't protect themselves. But they go behind leaded glass, they wear lead aprons, and they stay in their offices while other people take the x-rays (who also protect themselves). No risk whatsoever.