I don't own a tow dolly, I 4 wheel down. However, over my many years of moves in the military, and after I have rented a U-haul dolly to tow my car behind the rental truck I drive with my "stuff" numerous times (I've done this in FL, IL, CA, and VA). Never was I required to hook up the car lights to the truck in addition to the dolly lights.
U-Haul knows how to get blood from a stone, so if it was required (why would they break the law and be liable for a law suit) or they could make it stick they would make you rent or buy a hook up for the car lights too! All this being said, I would always make sure all my lights were working before starting out each day and I would check the dolly bearing hubs for excessive heat when stopping for gas etc. I will agree extra lights mean extra safety but they don't necessarily mean they are required by law. Now, if a "lawyer" wants to come on here and "object" I will certainly listen.
CMSgt, USAF ret.
2012 Canyon Star 3856//2009 Scion xB