There is nothing persay wrong with logging time in N reg aircraft.
But to fly in the US and to be able to log the time as JAR time you have to satisfy both systems. So to conform with the FAA system you have to apply for and obtain a FAA PPL be it linked to your JAR license or stand alone FAA license.
So without a FAA license you could log dual with an instructor or solo under their supervision but I don't know how that side of things works under the FAA system. It would get confusing with medicals etc.
There are no light singles or MEP's under 5700kg which under the JAR system you can log co-pilot time. Co-pilot is exclusively an issue to do with multi-crew ops and although it is possible to log co-pilot time in single pilot aircraft it is only ever done inside a commercial organisation. Who have to jump through hoops with the CAA and have to have SOP's etc setup for such operations. I have never heard of it done with MEP's.