Before a decision can be made about coyotes, one must first define if you are referring to the animal in their native environment, or in the introduced. Coyotes made their home east of the Miss. because of the 'sport' of fox hunting in the 18th-19th centuries. As fox populations declined, settlers were aware of the coyote and its physical and relative behavorial similarities. The creature was brought back east, and released, establishing the breeding population.
The ecosystem here in the east is not adapted to the presence of the coyote, and they pose a threat to humans (as recently shown), domestic pets, and native wildlife (they eat turkey eggs, etc.). They are an ALIEN species, and have no 'right' to be here, just like the Snakehead or the Starling or the feral Pig. Kill 'em all.
Conversely, they have had a presence west of the Miss for thousands of years, and that ecosystem holds a niche for them. In that case, it is we who are the invasive species, and expect nature to bend to our will. Don't kill them unless they are rabid, or pose a legitimate threat of some kind like no fear of humans, etc.
for reference, I am a native american/native ecology historian, and have made numerous presentations on the impact of humans and non-native species on the north american continent.
http://sites.google.com/site/southeastnative/