A Personal Auto Policy covers You and your household relatives first, and scheduled vehicles second. The insurance policy covering the vehicle is primary (first to pay) regardless of who is driving. Insurance on the driver is secondary if coverage on the vehicle is inadequate or non-existent. (Unless you are an auto dealer and then things get more complicated)
So, if you are driving your grandfather's uninsured vehicle, your own Personal Auto Policy would provide YOU with liability coverage for damage you cause to other people or other people's property while operating that uninsured vehicle. It will provide no liability coverage for your grandfather and will not cover damage you might cause to his vehicle.
Your liability coverage will follow you while driving that car. Your Collision and Other Than Collision ("comprehensive") coverage will typically NOT transfer to someone else's car while you are driving that car.
The above statements apply for rare and occasional use of that vehicle. If your grandfather makes his car "available for your regular use", you may have no coverage at all available while driving it.
If you are just doing some repairs and taking a test drive or driving it to a repair shop, I wouldn't sweat it. If he wants you to drive it on a regular basis and make sure it is constantly kept in good running order, I think your grandfather needs to add it to his policy or buy a policy with it listed as a scheduled vehicle.
Got ahold of my agent... no go. Insurance follows the cars except for rentals.
Whoa, whoa, whoa....an insurance policy that covers you generally WILL NOT cover someone in your household driving the car unless they are listed on the policy. This is what keeps people from not listing their 16 year old son on the policy and letting him drive anyway, for instance.
...but check with your insurer.