If the local population is not vaccinated, I doubt they will be allowed to be out in circulation operating all the businesses that serve tourists. I believe the vaccination program at the destination is critical to it being open for tourists.
While hard to compare to an island, Madeira (part of Europe) recently announced fully vaccinated tourists are welcome and can skip the otherwise necessary PCR tests.
We are a long way from Sept. but think we will slowly see over the coming months locations that rely on tourism open up to those vaccinated regardless if their own population has been fully vaccinated.
Not suggesting Italy will do this in a month but within 4 months if they are up to 30-40% of their own adults vaccinated including all in higher risk categories ; yeah I can definitely see that happening.
No one is vaccinating children yet, plus not every adult will agree to take a vaccine, so we are a very long way from the entire population anywhere being fully vaccinated in the US or in Europe. The travel industry will not survive if every nation prohibited tourists until that happens.