Does learner driver need to be insured when supervised by fully insured driver?

The learner will need to have insurance for himself. Let's say you are fully insured and you fit the criteria to supervise a learner driver; this means that you are covered but it's not you who's going to be in control of the car, it's going to be the learner. Therefore he needs to have adequate insurance cover in place (this is a legal requirement for any driver) whether you have a provisional driving licence or a full UK licence.

On a side note, the person who accompanies the learner (the supervisor) doesn't need to be insured unless he's going to drive as well.