An FAA CFI can (for free) teach someone for an FAA certificate or rating anywhere in Europe in an N registered aircraft. The problem arises if the course requires some solo flying for which they will need a JAA licence, RTF etc etc etc.
Why only for free? To be a CFI under FAA you need a CPL. Teaching in an N reg aircraft for a FAA certificate with appropriate DofT permission can be remunerated with a FAA CPL.