If you buy and use an Italian SIM (no matter where you get the phone), you'll have an Italian phone number. To use the phone's native calling and texting features, you'll be using the Italian number - which means an international call or text when communicating with people who have US phone numbers/US phone service.
If you buy smart phones for them, then you can use apps like WhatsApp to text/call without using the phone's number - you'd use mobile data (or WiFi) instead and wouldn't be required to make international calls or texts.
But if your two traveling companions are used to flip phones, they may not find it simple to use a smart phone. Smart phones are quite intimidating to first-time users, at least to first-time users who have put off getting one this long perhaps for a reason. And using an app to call instead of just dialing or typing a number to text may be more than they can figure out in the span of a vacation.
One possibility for you is to buy yourself an Italian SIM for your Verizon phone (which should not be locked). Then you'll have an Italian phone number too. Get flip phones for your friends with Italian SIMs. Then they can continue to make calls and send/receive texts like they are used to doing on a flip phone, without using an app, and because you'll all have Italian phone numbers, you'll be making domestic calls/texts. If you need to receive voice calls yourself, there are ways to do that without using your Verizon service. Google Hangouts and Google Voice is one solution for communicating with landlines and regular cell phones via smart phone apps.
Alternately, get smart phones for your friends before you all leave for Italy. Setup WhatsApp and make sure they can use it. (You can use it on WiFI without a SIM card). That way, you won't have to find out when they get to Italy whether they can make it work or not.