Ya' know, I have never actually bought a used car from a private party, but I read about the private party used car tax exemption in ads and heard from friends about it.
I just googled it and the law reads, "Casual sales between individuals are not subject to the use tax."
http://www.azdor.gov/Portals/0/Brochure/610.pdf
I also found a post on a dealer forum in which they were arguing this exemption was an unfair disadvantage to dealers:
States that don't charge Sales Tax on Private Party Vehicle Sales - DealershipForum.com
Yet another google search took me to a Yahoo answers page in which someone asked what other states have this exemption and someone said Georgia. I have no idea how accurate that is.
So I guess this exemption is quite rare indeed.