There are differences in several states, some tax, some don't. Some are 'sales' tax, some 'use' tax. Some states don't tax 'private' sales but do tax those between a buyer and a business.
Best to find out now, best to pay it if it is due... they are Good at finding people and then it is not just tax, it is Tax, Penalty and Interest in some cases.
a) AOPA website has some info.
b) a Tax Attorney licensed and familiar in your state?