I just looked up the 'dog' laws in Missouri. The second officer doesn't know what he's talking about. You have the right to kill the dogs if they are killing OR chasing livestock. Here is a quote from the law:
"Dogs may be killed, when.
273.030. If any person shall discover any dog or dogs in the act of killing, wounding or chasing sheep in any portion of this state, or shall discover any dog or dogs under such circumstances as to satisfactorily show that such dog or dogs has or have been recently engaged in killing or chasing sheep or other domestic animal or animals, such person is authorized to immediately pursue and kill such dog or dogs; provided, however, that such dog or dogs shall not be killed in any enclosure belonging to or being in lawful possession of the owner of such dog or dogs.
(RSMo 1939 § 14536) "
Here is the link to the website where I found the information:
Missouri Dog Laws
From the way I read the law, you have the right to pursue the animals (off of your property if necessary) in order to kill them. The only place you can't kill them is if they make it back home, or if they are actually in the possession of the owners (even if they are not on the owner's property).
I don't like the idea of killing anything, but I would rather humanely kill dogs that were chasing my horses than wait and have the dogs injure and horribly kill my horses. There is nothing humane about the way that loose dogs kill or injure livestock.