Assume you have smart phones? You can still use them on WiFi for no charge. Or buy a SIM in one country and use it in the others (EU has mostly eliminated roaming fees within the EU). Your phones must be unlocked to use a European SIM, however. Verizon smart phones should not be locked; AT&T phones probably are but can be unlocked by AT&T for free if they are paid off.
What would you want to use your phones for? To me, the most useful thing about a smart phone in Europe is navigation. Google Maps (and other apps) make it easy to get around by walking, public transportation, or driving with a smart phone. I used to struggle with paper maps and bus schedules before I got a smartphone, and mostly not needing to do that anymore has been a huge benefit to me while traveling.
If you want to make local calls within Europe, you might really want a local SIM card. I really don't need to do that when I travel, and mostly just want data.
If your Verizon or AT&T phones have Wi-Fi calling, you can also use them for free to make or receive calls home on Wi-Fi, without using them on local mobile networks. You can also use the free Google Hangouts app to make free calls home to the US, even to landlines.
You could also switch to Sprint or T-Mobile and get free unlimited roaming data and $0.20 per minute calls with their included roaming plans. Sprint, in fact, has a promotion right now where they will give you one-year free unlimited service for switching to them, if you have a qualified phone. I switched a few months ago, and it has been fantastic.