Actually Greece has a printing press for Euros (apparently you can tell which euro notes are printed in Greece by the serial number). I have been half expecting Greece to solve all its problems by just printing.
https://www.theguardian.com/world/2015/jul/06/return-drachma...
These were, of course, indistinguishable from real notes. The scheme ultimately failed when a mistake caused the printing of duplicate serial numbers already ordered by the state.
I doubt a scheme like this would work today in the slightest. If in an un-digitized world this didn't last long, in today's world it would quickly be discovered.
In practice they could only start their own “unauthorised” printing if it was kept secret since if it leaked the ECB would just issue new notes. This would likely be highly disruptive to the ECB as it was to Portugal.