No permanent stay permit or registration on "Anagrafe" needed?You can open a personal bank account in most of the banks in Italy, if you are a EU citizen. The passport and an address is enough to open it.
Bank might ask For an Italian Tax nr -Codice Fiscale. A lawyer or Account can help getting one
When did you do it and did they speak English at the counter? Even the webpages are in Italian only (yes, Volksbank has German version )Better you call them before to arrange an appointment and ask for their requirements.
I provided an Address in the EU.