Callers will only have to dial your normal home number in order to reach you in Europe. No "+1" prefixes are necessary unless that's also required when people call you in your home area. When you first switch your phone "on" in Europe, there will be a short delay while the phone registers which also tells your home network where the phone can be located. As the previous reply indicated, you will pay for each incoming call, usually billed in one-minute increments. If you're dialing from Europe back to North America, a "+1" prefix is usually required.
I assume you're travelling with a roaming plan with your home cell network, and that your phones have the two GSM frequency bands used in Europe? If your phones are "smartphones" such as iPhone or Android, you'll need to be VERY careful with data roaming, as the charges can be HUGE.
When travelling with your normal home number, it's important to tell all your contacts NOT to phone while you're in Europe (unless important) as it will cost you money each time (especially annoying if they're trivial calls) and it's also very annoying to get "wakeup calls" in the wee hours of the morning from some moron that forgot to check the time difference.