If a friend sells you their car can they keep their insurance on the car and if so can the new driver still be covered if not on the policy if the two parties verbally agree?

NO. Auto Insurance is not transferable.

An auto insurance policy becomes null and void at the moment the car is sold irrespective of whether the title has been changed or not. the seller has no authority to extend his policy coverage to the new owner. The policy contract prohibits this kind of activity.

You can not assume or acquire the Liability policy of another. If you are not a named insured on the policy then you have no coverage under that policy for your newly acquired vehicle. You must purchase your own coverage.