Yeah, if one is driving anything over 10,001 lb GCWR or GVWR for commercial reasons, the Federal Motor Carrier Safety Administration rules kick in (which all states have to subscribe to), and that is where the CDL game comes in. You can pull a 13,000 lb 5th wheel RV for pleasure without one, but if you are pulling one from the factory to the dealer, you are required to have a CDL, medical card (which is now computerized as part of the CDL), and log book (if over 100 air mile radius). Even the DOT has gotten on guys just moving their tractor and implement via roadway from one field to the other, because it crossed state lines, for not having a CDL. Yep, in a farm tractor pulling a planter or even a driving their combine! Goofy world we live in. As for personal stuff it is primarily left up to the individual states as to how and when they might require a CDL.
And any LEO that is doing weight checks and such regarding GVWR and GCWR, it is easy for them to run your VIN and check online right out of their car. Remember, you gave them your license, registration, and insurance card when they stopped you. They have all they need to see if you are legal.
True, they are not going to stop everyone. They are looking for key things as a reason to stop someone. Something looks out of place, you can bet their curiosity is going to be piqued and they are going to want to check. A loose chain and binder, a low tire, or any of a host of reasons. I operate a commercial semi truck, over 15 states, and I haven't been stopped and checked for almost 3 years. Not once since I bought my present semi truck in August of 2012. Crossed the scales and had weight checked, but that is it. Not one time in 3 years asked for paperwork, license, anything.