The local prepaid simcards will always be cheaper, but it may be more convenient to use your own if the costs are not exorbitant if family members need to be able to contact you for emergencies etc. However, you could give them your local number. You could carry both?
If you use your own, make sure your handset works on the Italian phone networks as the US has a different system. This usually means you will need a triband phone. almost 7 years ago