When I lived in Ohio, when you went to register a car that you'd bought private sale, they looked up the book value, and if you were some percentage below that, you had a choice of paying tax on the book value, or getting sworn notarized forms that this was the price you really paid, and if you were caught lying, you could go to jail.
The one exception was that if you bought a car from a relative, they accepted that you might really have paid a below-market price.
I don't know how vigorously they investigated, maybe it's an easy crime to get away with. If they really cared, then if you paid with a check they could surely easily subpoena records from the bank to get the amount of the check. Even if you paid with cash, if they see that the buyer withdrew $X from the bank the day before the sale and the seller deposited $X the day after the sale, that would be strong evidence that that was the sale price. All things considered, I've never thought that saving a hundred bucks or whatever on the sales tax was worth risking going to jail.
And as to "will I get in trouble as it was the buyer's idea?" ... think about that for a few minutes. "My buddy Frank suggested that he and I rob a bank, so I went along, I shot the guard and drove the getaway car while Frank was the one who grabbed the money. If we get caught, will I get in trouble given that it was all Frank's idea?" Yes, you will.