Guys the responses on this thread may not be valid. The laws on the tax on cars vary state to state.
In some states, like Georgia, there is no sales tax on used cars purchased from a private party. Only sales tax on used and new cars purchased from car dealers. So in Georgia what the OP says is correct. In Florida and Texas, there is sales tax on used cars purchased from private parties. So clearly different states have different tax policies.
I am not familiar with Arizona law.
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