Not sure about New Jersey, but in Arkansas you already paid income taxes on the money when you bought the car (think about it -- you didn't get a deduction for buying the car, did you??). So here there are no taxes the seller pays.
In some states there is a tax due to the state from the buyer if they have a state tax. Other states no state tax like NH charge a fee to to register the car which is expensive. But no fed tax is due on a private person to person sale. So you don't owe any money to state or fed.
