Only the CFI of an approved modular school in the USA has to hold a JAA licence and instructor rating provided that the ratio of non-JAA to JAA licenced instructors does not exceed 6:1 for PPL training and 4:1 for CPL/IR/MEP/FI(A). There is no limit to the ratio at schools approved for integrated training.
The FAA refer to the process whereby a pilot certificate is issued on the basis of a foreign licence as 'conversion', but then they never were very good at the English language. They will 'convert' any ICAO licence but will only allow private privileges as detailed in FAR 61.75(a).
A PPL is sufficient for an instructor within the JAA only if no remuneration is given. To be eligible for a FAA flight instructor certificate, a pilot must hold at least a CPL and Instrument Rating as detailed in FAR 61.183(c)