I may need meds refills while in Rome. I've been told it's not that hard.
Someone told you wrong. Once upon a time, getting prescription medication over the counter was fairly easy (depending on the the pharmacist). No longer. European Union rules and a new generation of "fiscal" pharmacists — those who actually pay attention and stick to rules — now reign in much of Italy. If you don't have an Italian Rx, good luck.
If you're in a hotel, it means getting the concierge to get a doctor (in higher end places) or relying on an outsider at a high price called in by a pensione or friends of the pensione owner. Bottom line, you can of course visit an emergency room (pronto soccorso, but that's really not advisable unless there's a real emergency.
Sure, you can always beg. Pharmacists, if they understand the problem, do have dispensing power. Rome has an international pharmacy (Piazza Barberini) that will at least take a look at your North American Rx.
The best advice is to come equipped with what you need. If the bottom falls out, medico is the word for doctor and emergency in English has an international ring. If that gets nowhere, turn to an English-language doctors. They usually have their mobile phones handy. Be advised your "emergency" Rx won't come cheaply. If a doctor is actually dispatched you're talking a minimum of $150-200 (€100+). There's really no such thing as a doctor calling in a prescription in Italy, since they're all still done by hand and pharmacies don't have computer databases.
Another tidbit to remember is that prescriptions for anti-depressants, sleeping pills and their ilk last only 30 days from the date written down by the physician. After that, the Rx is worthless.
Here are a useful few numbers for Rome: International Medical Center, tel. 06.488.2371; 24/7 service (English spoken); MEDI-CALL, tel. 06.884.0113; Rome American Hospital, tel. 06.22551; MD on duty 24 hours; Salvator Mundi International Hospital, tel. 06.588961.
Regarding dentists, the U.S. Embassy recommends you call G. Eastman Dental Hospital - tel. 06.844.831, with 24-hour service. The International Medical Center (see above) also offers 24/7 dental service.
Final note: Are these numbers foolproof and 24-hour medical/dental service always assured? No. But Italy is fast catching up to European norms.