Properly fitted bumpers are a snug fit to the car, maybe an 1/8" free, or less, on each side. And, you are right, they have to be encouraged into place. I had old bumpers refinished and part of the process was arcing the bumper to suit. First time it was a bit loose and I had them tighten it up by a 1/4". Once to the correct size, then they did the polishing and plating.
As to fitting, the bumper is more "open" underneath. I draped rags on the body at either side, dropped one side in place and carefully slid the other side down and into place. There will be some tension but the soft rags will slide with the bumper and you paint will be fine.