I have much the same situation. I'm at the end d of a 1/2 mile private road. The township used to snowplow it, but quit doing it. My neighbor (also at the end of the road) has a truck and a plow. He didn't want to be 100% obligated (i.e. responsible if he was out of town), but does a good job. I pass out letters and envelopes asking the neighbors of $25 - $35 each (for the whole season) to cover his expenses. Initially, it took a couple of attempts with some, but they have come to realize what a bargain it is. I pay the fee for the widow. In big storms, I help do the driveways of those who don't seem to have a plan.
In the spring through fall, I maintain the driveway with a pony grader and a road drag. I used to do that for free, but after having to do an extensive rebuild of the drag, I asked for $25 donations. I said I wouldn't keep any of the money, but instead use it to buy more gravel for the road. After the first year, it went from donations to an annual spring "bill". I now have them pretty well trained. My requests have everyone's name on them with their name circled, so they can see everyone is pitching in.
Before I did either, I talked with about 3/4 of the neighbors and told them what I was thinking. And yes, they were the ones I thought would most likely go along with the plan. Peer pressure and common sense took care of the rest.
I have good neighbors, but human nature still bemuses me. It is surprising how many times a neighbor will drive around a blown tree, somehow expecting it will magically be removed :>)