Should good prosciutto be refrigerated?
Mortadella does need to be refrigerated.

As for the proscuitto ... if it's real cured ham (and not the fake 'cured' stuff you get in the US) and was trimmed with a clean knife and wasn't otherwise contaminated when being cut, and it's not the middle of the summer where you are, it's likely fine being stored in a cool area of your kitchen.

Mind you, the US FDA would't agree, and you would get cited for health code violations if you were a restaurant.  You may also want to be more cautious if you're going to be serving the meat to people with compromised or otherwise weak immune systems (eg, small children or the infirm).