Each state has its rules. Some require formal training and some don't.
Some states that require the training will pay for it and others make the rider pay.
Amazing. Here in Spain despite having had my car driving license for more than ten years, I still had to do three official exams to get a "limited A2 motorbike" license, which still only allows for bikes up to 35 kW (47 HP). The exams were:
1) a written test about all things motorcycle (20 questions, and you can fail no more than two);
2) a riding test on a closed circuit, with two parts, A) to slowly sort through some cones very closely spaced among themselves and placed between two white lines which you cannot touch, without any time limit, you cannot set any foot on the floor; and B) to do a speed circuit going through some fast cones and doing a 180º degree turn and then making an emergency brake at the end, in less than 25 seconds.
3) a riding test in the open streets, while followed by a car with the examiner, wearing a helmet fitted with an intercom to receive instructions about which way to go, it lasts about 15 minutes and you have to nail every Stop and every pedestrian crossing and every red light and every yield, otherwise you are toast.
And then the examiners are not any private motorcycle instructors you can hire, but public servants unknown to you.
I was lucky and passed all three exams in the first try. I still had to pay about 550 Euros for the whole fiesta: to gain access to a private facility replicating the slow and fast circuits to train for those tests, and to do a mock riding test in the open streets with a private motorcycle instructor (useful because the private instructor here warns you about common pitfalls the examiner is going to try to make you fail), and to pay for the three official exam taxes. If you fail any of the three tests, you have to take the test again and pay again the taxes for the failed test (I think you are required to wait a minimum of 15 days if you have to retake any test, not sure as it wasn't my case).
And then, after two years holding the A2 license, I will be eligible to pass another riding exam with a bigger bike if I want to upgrade to a full A motorbike license. This will be about 300 Euros additionally, because it legally requires also sitting through 8 hours of in-situ lectures about motorbike safety by an approved private motorcycle instructor (no official test needed afterwards on that theory, thankfully).
Welcome to a nanny State!
Here you can see a famous Youtube video about practicing the slow and fast riding tests on closed circuit for the A2 license (warning: it's Spanish spoken):