I've been around them all my life. I remember it being spooky walking home as a teenager, late at night, long dirt road and the coyotes whooping it up somewhere on the other side of the fence that separated the field from the road. They always sound like so many and so close.
Now a days, I often see their tracks around my house in the mud and snow. In the winter they are easy to spot, traveling on the lake.
The only time we ever have problems with coyotes or any other animal for that matter, is when people start feeding them. Otherwise they just keep the critters in balance like they are supposed to.
That's my feeling too. We have them around, I think maybe the road out front is a territory boundary lately. I will hear two different groups howling and yipping, one from each direction. They leave scat on our trails and our dogs have scent marking contests with them.
I know feelings about coyotes run strong. They are an invasive species here like most of the eastern US, but most likely only because wolfs were eliminated leaving a predator niche open. I don't like it when deer hunting revenue becomes more important than wildlife diversity and balance. Besides the research showing that coyotes increase their litter supply to match the resources available, it just isn't a valid long-term strategy to manage wildlife to optimize for one species such as deer.
When deer hunters pop out of the woods 100 feet from my house, it means they don't even know where my house is. That is not a comforting thought and they represent a bigger danger to me than any coyote ever will.
We certainly have coyotes around, but also deer and the neighbor's roaming cats. I can't see that having coyotes is eliminating the deer by any means and the cats are surviving too. To my mind, the cats are probably harder on native wildlife than the coyotes when cats are allowed to roam. The cats don't belong here either, they are also an invasive species. When coyotes eat cats, it could be because the cats aren't where they should be to begin with.
The hair in coyote scat here looks to be mostly snowshoe hare, in the late summer it is loaded with blackberry seeds. I'm sure they eat their share of white-footed mice and voles too.
The people who keep livestock may have a valid reason to kill coyotes but even then, somewhere there has to be a balance between protecting livestock and decimating whatever happens to make that more difficult or expensive.
The woods around here would be a poorer place if I can't step outside and hear a coyote now and then. As people take over more and more habitat space, we have to make choices on co-existing with and preserving wildlife if there is to be any left.