You are right about Sudtirol special law but the standard law about hunting permit is the same in fact if someone the hunting license from the Bozen office, he can hunt on every private area and some special public area of Italy.
About hunting tradions in Italy, there are mainly in Sudtirol and some alps places(Valle d'aosta, Piemonte and Friuli venzia Giulia) for the ungulates such as Roebuck, Red deer and Chamois and in Tuscany especially the Maremma for the wild boar driven hunt.
PS Sudtirol is not Italy only for someone and for the maps, it is Italian.