Until very recently, Utah had a law on the books that not only prohibited legal marriage to more than one person, it also prohibited LIVING WITH more than one person whether you were married or not.
The Browns faced prosecution because even though Kody is only legally married to his first wife, Meri, he was in effect cohabitating with all four of his wives, thereby breaking the law.
In large part because of work by the Browns, that law has now been changed; while Utah, like (as far as I know) every other US state, still prohibits *legal* marriage to more than one, they no longer have a law against cohabitation with more than one partner.
As far as I know, in all other states, the only law is against legally marrying more than one person; living together and even having non-legal commitment ceremonies with more than one person are not against the law. (Best Friend has a high school friend who recently joined with two men in a single commitment ceremony in New York, for example.)