The key thing to know/remember, is that your account with your home (USA) carrier is entirely within your SIM card, not your phone. So, for example, if you were visiting me here in Canada, I could put my SIM card into your phone, and it instantly becomes "my phone" with my number. Ring or text my number, and your phone receives it because of my SIM card inside.

Ditto for Greece. Get a local SIM card there, insert into phone, and the phone is now entirely a "Greek" phone, with a Greek number, on a Greek network. Except it still has all of your apps, preferences, and customizations intact.

Similarly, you could insert your USA SIM card into a phone of a Greek host, and make calls on it -- billed to your USA account.

So the account is with the SIM card.