It's way too early to know. As with most things related to the pandemic in Italy, 99.9% of the information is targeted to residents in Italy returning into the country. I'm vaccinated here in the UK and right now there isn't a whole lot of information on how to recognise the UK vaccinations and viceversa, despite both countries having a good back-end system with online registers and so on.
My hunch is that, for the Americans, things will continue as they are today. Sure it'd be great if somebody in the US had asked Jeff or Bill for some space in AWS or Azure to build a database of vaccinations with an API in front but hey... warp speed!