I do carry a gun but sometimes I loose sight of them.
I do a lot of off-leash hiking with my dogs on our property (74 acres) and our neighbor's adjoining property (105 acres, with neighbor's permission) and I NEVER loose sight of my dogs. They don't get to be so far away from me that I cannot see them - that's a safety issue. IMHO it's also a training issue - they should be close enough to where I can see them.
In my area, we have A LOT of coyotes. They do bother our neighbor a lot more than they bother us, primarily because our neighbor raises and keeps wool sheep. He has, on several occasions, caught coyotes right inside his barn, or outside in his sheep pasture around dawn. That seems to be when they are most active.
Early fall last year we had a big issue with a female who had pups and would actually come out in the middle of the day. When the neighbor was haying one of his fields, she was out in the field, catching/eating mice that were chased up by the tractor. Middle of the day with people working in the field.
We started having much less of a coyote issue when we started hunting them on request of our neighbor after they'd killed several of his sheep within a short period of time. Once we started shooting them, they moved on, although we still sometimes hear them at night, though not nearly as close as they used to come.
Our coyotes here are approximately the size of my Malinois and very similar in coloring to my Shepherd. As they're known to attack sheep - which are taller and heavier than either of my dogs - I don't think they'd shy away from attacking "large" dogs. Actually, there was a case here in NY this summer where a pack of coyotes attacked a young girl (I believe she was 6 years old?) in front of her house!