What activities are the Eurozone central banks permitted to do?

[plaguedbyfoibles]

They can't engage in open market operations (unlike the ECB), they can only print Euro notes with the authorisation of the ECB, they implement ECB policy decisions, partake in ECB elections etc.

[csearle]

Well the French "central" bank has three main missions which are...

...to drive the French monetary strategy, ensure financial stability and provide services to households, small and medium businesses and the French state.

Of course being in the EU means the bank doesn't decide the stuff that actually matters, that is done for them. C.