To address the basic question, the laws in some states require brakes on any trailer over 1500 lbs. unladen weight. A dolly weighs considerably less than 1500 lbs., so in those states no brakes is perfectly legal.
In some states it is questionable if a dolly is legally a "trailer", according to the statutes. If it is not a "trailer" under the law, it is not required to comply with the laws regarding trailer brakes. In those states, a dolly probably does not have to be titled or registered, and does not require a license plate.
In some states, any trailer over 3000 lbs. GVWR must have brakes, often on all wheels. Again, there are often exemptions to the law.
So, depending on the trailer braking laws in each state, a dolly without brakes may be perfectly legal, until the Braking Performance Laws come into play. Most states, and the FMVSS, have a braking performance law or regulation that states any combination of vehicles must be able to stop within a set distance (usually 40 or 45 feet) from 20 MPH on a dry, hard, clean, level surface. If the combination of vehicles can do that, it is legal. If it con't do that, better braking is required before it can operate legally. Here in Montana, that law is MCA 61-9-312.
I have never heard of any testing for compliance being done, nor have I ever read or heard about any citations for violations of the Braking Performance Law.