I am actually curious how you would go about proving that you are a citizen of the EU. I am sure you need some form of identification to board an international flight even though it might be within the Schengen area.
You'll need your ID to stay in a hotel/AirBnB though. So unless you squat or camp illegally, better have it with you.
Locals have been able to travel between the countries freely without a passport since 1954.