Insured to drive someone else's car, does that car need to be insured by that person as well?

This can be a tricky situation - if the car is not insured by the owner, most likely he is not using it. That probably means the car is not taxed or MOTed which would render the car unroadworthy. Now any insurance cover will become void if the car in which the accident happened does not have the legal documents to prove that it's roadworthy.

For peace of mind, call up your current insurer and ask them before you drive the vehicle.

No, you can't drive the car unless the car itself is insured. Your insurance allows you to drive another car with the owners permission but that owner has to have insurance on it.