You're not going tolike this - but it seems to depend on the geographical location of where you want to teach.
I know some JAA guys have been teaching nothing without FAA qualifications, I know some who have taught anything that doesn't require a "sign off", I know some who have taught everything (even first solo).
Some have letters of dispensation from offices of the FAA, some don't.
No matter what rule and legislation is quoted to you in any following messages - IN PRACTICE it depends where you are.
Without doubt, best bet is to get both ratings (and a VALID visa for working, of course[which does NOT include the M1 student visa issued by some schools]).