hc4361 - maybe they could, but if you are driving my car under your policy, your policy will not pay out if the car sets on fire or is stolen from you. I'm not certain whether mine would or not. It might have to go to "pistols at dawn" between me and you.
For the first time in a long driving career using Fully Comp I was surprised that Admiral will not give me the entitlement to drive other vehicles (3rd Party) on my current Fully Comp insurance just because of the vehicle I drive for SDP. I have a Nissan Nivara.
It's not something I've seriously looked into for my own policy. The only situation I can readily think of for me driving someone else's car is to consider buying it from them or trying to diagnose a fault. In either case it would be just "round the block" with the owner sat in with me and the car never left unattended, so theft unlikely and, if it caught fire, that's their problem.
More important than Fire & Theft not being covered, if you have an accident which is your fault the damage to the car you are driving will not be covered.
There is no automatic cover just because it's comp. If it says you are able to drive other vehicles with the owners permission on your certificate then you can, usually third party only. If it does not say it then you cannot. This facility is intended to make you legal to borrow a friends car for example but as others have pointed out you will not be covered for theft/fire and indeed damage to the car you are driving.
The removal of DOC cover from many insurance policies stemmed from increasing abuse of the facility. Originally provided to cover the odd emergency situation, it has been increasingly misused.
Furthermore, if you look at insurance certificates that do give DOC cover, you'll see that the cover excludes collecting vehicles that have been impounded for no-insurance offences. Uninsured drivers were getting their mates to collect their vehicles using their mates' DOC cover!