Euro banknotes are serialized with a letter from the latin alphabet and an eleven digit number as in: Y12345678910. Each country member of the EMU is currently assigned a fixed letter. “Y” is for Greece and “X” is for Germany. Moreover the 11 digit number is constraint by a “checksum”. The remainder of its division by 9 has to be fixed. Greece’s checksum is 1 and Germany’s is 2 for example. So Y12345678910 is a good Greek euro serial number because 12345678910 divided by 9 turns a remainder of 1. Y12345678911 is not a valid serial number for a Greek Euro. So in principle you can tell which euro belongs to which country. For more on this arcane stuff see here.
Each country owns plates and facilities for printing Euro banknotes but they need to behave themselves and coordinate with the ECB before they do so. The theoretical question of which army would stop a rogue country from printing Euro banknotes as it sees fit without coordinating with the ECB first is in fact not so theoretical, that’s why the good people at the Deutsche Bank came up with the Geuro. I’d say take a further step and give each country a second letter. So for example give Greece X and Germany Y. Keeping the checksum fixed will still allow you to know who printed the banknote if need be. In fact with the latin alphabet having 26 letters (i.e. 25.26/2 = 325 pairs) and the 9 possible remainders you could give each of 2925 countries two types of serial strings. When Germany feels like having the DMARK they would use the Y instead of the normal X and whenever the Greeks get in trouble they should use the X instead of their Y. Spain has V and checksum of 4. I would propose that they get the X and keep the checksum fixed. An iPhone app would allow the layperson to scan and check both authenticity and origin. Whenever you feel like depreciating you can print some of your secondary lettered banknotes and everybody will know what you’ve done. Drachma and DMark anyone?