It depends on the age of the car, and what happened to cause the bearing to go bad.

When I had my 2013 Fusion, I hit a raised manhole with 5 people and my own luggage in the car while visiting relatives. The country road had been ground down and I didn't catch the manhole until it was too late. It was dusk and the road was shaded by trees. I ran over the manhole going 55 MPH. Nothing happened right away. A few hundred miles later, I started noticing a rumbling when I drove home on the highway. Left rear wheel bearing was damaged. I replaced the whole hub a few weeks later and didn't mess with the others.

In something with high mileage, or is very old, I would probably replace both sides if one were going bad. 

You can just do one, but if one has worn out the bearing on the other side is probably not far behind.

One at a time is fine, Only thing I would add to this is if you’re replacing the bearings on an axle that requires opening up the differential and pulling an axle shaft out you might as well do both sides especially if it’s an older vehicle. The bearings are relatively inexpensive and since you already have the diff open and  access to the c clips to pull the axle it wouldn’t hurt to do the other side while everything is accessible. 

In my experience they can fail many years apart. Besides, you don't save any money by doing at the same time.

Just do one.

One at a time is perfectly fine.

It’s not like the labour is cheaper if you do them in pairs.