We applaud you and your friends for having a designated driver when going out and drinking. It is a wise choice to have someone responsible to not drink and make sure the others make it home safely. When you are the designated driver and not going to be driving your own vehicle than the owner of the vehicle should read through their policy and even contact their insurance company to make sure you will be covered if you were to be in an accident while operating their car.
In general most personal auto insurance policies extend coverage to others that the car owner / policyholder gives permission to operate their vehicle. This is known as a permissive operator or driver. There are however some car insurance policies that have exclusions to the policy which may exclude you from being covered as a driver. Exclusions can involve other drivers entirely, drivers under a certain age, etc. That is why it is best for a person to read the terms of their policy and double check with their insurance agent that you or others are allowed to drive their car and have their coverages follow and extend to you as a driver. As a general rule, you would be covered if you are given permission to drive the vehicle.
If you have your own insurance policy than it typically would be secondary to the car owner's policy. So in you were at-fault in your friend's vehicle their insurance would pay out first and if their liability limits were exceeded than your insurance policy may be used as excess.
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