@Willow2253 focus along this line, not the drama. Do not get side tracked by the "what ifs" and comparisons with loss to wildlife. Now check your county statutes.Here you go. To find out how to report it in your county, search "(name of county) dog kill livestock" and you should find some guidance. In my county there's a dog incident form to be filled out.
As for Oregon-wide laws, this is what you're entitled to. The page lists other applicable laws.
https://www.oregonlaws.org/ors/609.140
It says:
(1) The owner of any livestock which has been damaged by being injured, chased, wounded or killed by any dog shall have a cause of action against the owner of such dog for the damages resulting therefrom, including double the value of any livestock killed and double the amount of any damage to the livestock.
(2) If one or more of several dogs owned by different persons participate in damaging any livestock, the owners of the respective dogs shall be jointly and severally liable under this section. The owners of dogs jointly or severally liable under this section have a right of contribution among themselves. The right exists only in favor of an owner who has paid more than the pro rata share of the owner, determined by dividing the total damage by the number of dogs involved, of the common liability, and the total recovery of the owner is limited to the amount paid by the owner in excess of the pro rata share of the owner.
(3) An action brought under this section may be tried as an action at law in any court of competent jurisdiction.
(4) As used in this section:
(a) “Owner” means the head of the family of the home where the dog is cared for at the time of the damage.
(b) “Head of the family” means any person who has charge or manages the affairs of a collective body of persons residing together, the relations between whom are of a permanent and domestic character. [Amended by 1973 c.655 §7; 1975 c.749 §1]
In the end you must work to prevent repeats involving dogs in general. More likely than not, that will require some form of fencing. Still try to get money for losses incurred, but do it in a calm and structured manner. If dogs come back, then shoot them to kill. But keep level head.
I have gone through your situation myself more than once. You need to start working on a reputation, one that is positive with your neighbors as a whole. Be one, not an outsider seeing faults in the local good old boys.