Here it is state by state. Some states have little in the way of regulation, some have a lot, but few have much in the way of enforcement. For my state, Massachusetts, Manicurist-Type 3 license, you have to have a month long, 100 hours+ training program and pass a 90 minutes written exam (offered in English, Spanish, Mandarin Chinese, Portuguese, or Vietnamese), plus a 90 minute practical exam. Additionally, we separately license salons as well, so to set up shop here, you need both your manicurist license and a salon license or work for someone who has the salon license.
There are few products that an DIYer can't get since there is no statute regulating products. There are a lot of fake products out there, especially OPI, so it's easy for a DIYer to end up with counterfeit products. Some stores, like SalonCentric or CosmoProf, you can't even enter without showing your license, some will only sell you certain things without a license. I only buy from authorized sellers. My license expired about 8 years ago and I still go to the same shop I've always gone to with no problems (I use mostly OPI GelColours and CND products, with an occasional LeChat, Cuccio, or Gelish thrown in); perhaps it's because they know me, but I've never seen them ask anyone for their license.