And they each need to be to a full stop and in proper night (i.e. 1 hr after sunset to 1 hr before sunrise).
As a summary
For a standalone FAA and JAA licence you must have
BFR within the past two years (FAA CFI)
CofE or CofT review within the past 24 months (JAA CFI)
An FAA medical
A JAA medical
For a piggyback issue you can use the medical from the same country as the licence on which your FAA is piggybacked.
As IO says, it is not clear what happens when you then add new ratings (like the IR) rather than convert say a JAA IR to an FAA along with the basic piggyback.
FAA licence allows you to fly Nreg anywhere
N-reg can also be flown in a country by a pilot with a valid licence of THAT country and a valid medical of That country. (Note just because licences are JAA does not make France and the UK the same country as far as the FAA is concerned)
There is an accident report of an N reg Kingair in Ireland (AAIU site) that shows how you can end up not being legal for the flight by getting the wrong combination of paper from different states.