when turning on the rear fog light, the driver's side is the only one that illuminates. is this how it's suppose to properly function?

 

Yes. Only the street side is illuminated. When I lived in Germany a long time ago, people would mount a white light on the rear corner of their car as a rear fog light.

 

White lights in the rear are illegal in the U.S. when driving. The brighter red light during foggy weather helps motorists behind you to see your vehicle from farther away. By lighting up only one side it doesn't confuse them into thinking you are applying your brakes.