Have one of the rigid wood-look chair mats on carpet, a Costco purchase. Despite the fact that it has a slightly textured back, it still moves around as @Axtremus has described. The other thing that I find to be a pain is that the chair rolls around too easily and I feel like the slightest movement makes the chair move when I don't want it to.
We used to have a clear, slightly flexible plastic/polycarb chair mat with gripper teeth on the back. It had a little more texture on the top so the chair didn't "float" around as easily, and those gripper teeth were very effective. And very pointy, almost like a thorn. A bitch to set up, but once it was in place, it didn't budge. It lasted a long time (years and years), but because it was not rigid, it would flex downward a little into the carpet and it got tons of little cracks from a person sitting in the chair and moving it around. Eventually it cracked all the way through and I replaced it with the one we have.
The best arrangement is to have a rolling chair on commercial carpet that isn't pile but rather looped. You get a little resistance with respect to rolling, but it's still much easier than moving on a thicker carpet.