You will have noticed that when the actual x-ray is taken the radiographer comes behing the screen? The screen is lead lined and the glass is lead glass. There is no real need to waer a lead-coat behind the screen.
Radiographers are NOT in daily contact with harmful radiation. An X-Ray machine produces radiation only for the duration of time of the exposure. It is not a source of radioactivity like an isotope but rather produces photons (no star-trek here!) which are the x-rays by bombarding a tungsten target with electrons. The actual exposure is incredibly short- for a chest x-ray for instance it can be as short as 10 milli seconds.
When the x-ray beam is produced- within the x-ray tube- it is emitted through a port in the side of the tube- so it only goes in one direction. Beneath the port is a thing called a collimator- you may notice that when they set up for an x-ray they shine a light onto the area to be examined. The light indicates where the x-ray beam will hit- and the size of the area to be irradiated can be made larger or smaller by moving lead shutters within the collimator. The amount of radiation outside this area can be measured as negligible.
I fix x-ray machines for a living- like radiographers I am required to wear a dose monitor at all times when working. Health and safety is VERY strict when it comes to radiation. My does badge comes back from scrutiny every three months and is always a zero reading. If it were any different I would have some explaining to do and if a large dose were measured I would not be allowed to work!