The issue of who should weigh is often confusing because many states operate with old and sometimes unclear statutes or language and even more unclear signage. Then some do not enforce their laws the way the signs seem to indicate.
icman 95 said it pretty well above. Everyone driving a truck and pulling a trailer, even privately owned and not engaged in commerce is subject to the laws.
Private owners driving a pickup pulling a recreational TT or 5W, both registered to the driver, are generally not expected or required to stop at scales.
California scale operators or CHP will come out and yell at a commercial hauler pulling trailers for hire with DOT signs displayed and wave you off their scales if you are driving a pickup truck with a pickup bed but require you to stop if you are doing the exact same thing with a flat bed or no bed at all.
Some states have signs out that say "all trucks must enter scales" and some say "... ... ... ... ..... all commercial vehicles. " Other states use signs stating "all vehicles over 16,000 lbs. must weigh" or "all vehicles over 8 tons... ... " and others vary with weights like over 20k or over 26,000 lbs.
Then there is the matter of interpretation and enforcement. Some troopers or scale operators are busier than others and may actually enforce a policy less strict than their signs indicate. During the period I was hauling RVs commercially I found that the signs and expectations of law enforcement was often unclear but I was expected to know what was required. There were some states where I always drove right by their scales even though I was loaded and pulling for hire and others I knew better than to even think of driving by because they'll chase you down and fine you heavily. Experienced OTR drivers know all these requirements from long experience so DOT often expects everyone else to know as well.
Generally speaking if you are driving your own pickup, pulling a recreational vehicle (not cargo trailer) registered to you, you can probably drive by the scales without being bothered. But if you are driving a truck that looks commercial, for example one that has a flatbed or utility body installed, it might be wise to cross the scales and give the scale operator the opportunity to look you over and either wave you on or direct you to come in with documents to prove you are not commercial. There are some scales in some states that are so busy with big trucks that the troopers or scale operators will get angry and come out and yell and wave their arms at you telling you to get the hell off their scales and out of the way of the big trucks.
I guess the short answer to your question is, "who knows?" It's not always clear.