This could turn out to be a long thread but let's start with this.
Broadly speaking yes we can!
For training towards the issue of a new licence or qualification the instructor, the aeroplane and the organisation all need to be authorised. But, to teach a Cessna pilot to fly a Piper, a runway pilot to use grass airstrips, a domestic pilot to fly to a foreign country, an old fashioned guy to use a GPS, (a modern pilot to use a map!) etc etc you don't.
Happy landings!