Agree with trapperdave. Coyotes usually go berserk in a snare, and sometimes hang themselves if there's enough brush or trees nearby. But Fido usually just lays down and quietly gives up.
BTW other than a hunting dog on a hunt, its illegal to allow pet dogs to roam free, anywhere & anytime. So if you catch someone's pet, the owner of the pet is in large part to blame.
.