When I got my license, my dad called the insurance company. They said they didn't have to add me till I had my own car, got a ticket, got in an accident, or turned 18. So my cost to my parents was $0. That was Citizens insurance. From what I hear from friends/coworkers, when their kids got their license, the insurance company automatically got notified and was calling them up immediately. My understanding is that a lot of the discount insurance companies will not allow you to have kids living at home with a drivers license and not put them on the insurance, UNLESS you list them as an excluded driver. In other words, if they ever drive your car, you have $0 coverage. Zip. Nadda.
My son is 2, so I've got a long ways till I have to deal with this, but I know I certainly am not going to foot the bill for $2k a year. If he wants to drive our car, he can get a job and pay it off. As to learning how to drive, do you have to add your children when they are just on their permit? If not, maybe leave them on their learners permit for the maximum duration. Sure, they can't save you time by doing errands for you, but at least they get experience driving. For me, I would run a lot of errands for $2000 in savings a month.