The general answer to this question is complicated. But the specific example makes it a lot simpler. First the complicated bit.
The Duchy of Toscana would appear on the map as part of Poland. All taxes, levies, etc. would go to the King of Poland. Since Toscana is part of the Kingdom of Italy, the King-Duke would be able to vote on certain Italian law-changes, successors to the King of Italy if Italy was Elective, and generally be involved in Italian politics in a way that was impossible before he inherited. It's similar to the situation of Normandy in France. His Italian vassals follow Italian law, so it's entirely possible he could have no Crown Authority in Poland, and thus no right to do anything, but have the right to revoke titles at will in Italy because they have high Crown Authority. This is all a pretty good simulation of the relationship between the English King-Dukes ofEngland-Normandy and the Kings of France.
Now the simple bit:
There is no Kingdom of Italy in 1066. Italy is not de jure part of the HRE. Which means there's nothing to vote on. Italian Crown Laws are what they are, and nobody has the authority to change them. There's no throne so you can't have an election electing somebody to it. Your Italian territories will act exactly like a Lithuanian Duchy you conquer on a northern Crusade.
I can't tell you whether these Duchies will follow your laws, or the laws of their official, uncreated, Kingdoms. The latter means no Crown Authority, virtually no tax money, no ability to take anything away from Nobles, no ability to change your military commanders, etc. The former means you'll have the same rights as you do in Poland. The preview apparently switched between the two methods, and I don;t have access to the demo myself (no Mac version) to try it out.
Nick