Hey there,
I am so, so sorry about your dogs. I realize it's been a while since you posted this but it's floated back up. I felt the need to contribute due to seeing some myths about dogs in here that I wanted to correct.
I've been using the Koehler Method of Dog Training since 2010 to help reform problem dogs. So that's why I feel my opinion is worth something.
If a reader doesn't want to read my whole post, as least read this: animal aggression is NOT the same as human aggression. prey aggression is NOT the same as dog aggression. While any dog may have one or a combination or all of these types of behavior, they are NOT a slippery slope that inevitably leads to another! Your dog knows the difference between a possum and a human baby. A dog may attack livestock, dogs, and humans, or they may only have an issue with other dogs, or only an issue with people. It completely depends on the individual dog and their motivations. Particularly with prey chasing and killing, this is a natural, normal dog behavior! We (rightly) tend not to assume that a dog that enjoys chasing squirrels is next going to attack your toddler. That's not how it works. If you have a high drive dog that has not been taught that children are not to be interacted with as prey, that's another story. But your dog that normally has great manners with children and then kills a chicken is not going to suddenly decide they'd like to try human flesh.
As you've already gathered I'm sure, it all depends on the dog. Some people here have gotten a whole lot of mileage out of working with their dogs to change how they view livestock. Personally I wouldn't ever leave a non-livestock guardian alone with livestock, ever - unless they'd been raised with the livestock since puppyhood and had zero red flags at a mature age. I would count "alone" as being outside together without a human present, since goats and dogs are great at overcoming the challenges of most fencing. If you're not there to remind them what they ought not to do, I wouldn't rely on them to make the right choice. Instinct is a powerful motivator - THE most powerful motivator.
Here is an article called "The Misbehavior of Organisms". It's long, but recommended reading for anyone interested in training any animal. The point enclosed is that despite teaching various species of animals to reliably perform a behavior for reward, eventually each species would cease to do the trained behavior in favor of doing an instinctual behavior that got them no tangible reward. (a pig did "rooting" behavior, a raccoon did "washing" behavior, and a chicken pecked incessantly at a small moving object) It illustrates how instinctual behavior is more rewarding to the animal than anything we could ever reward them with.
http://www.niu.edu/user/tj0dgw1/pdf/learning/breland.pdf
Which is not to say that animals can't be taught to abandon those behaviors, but that it is never going to be something that is totally fixed with a positive reinforcement only approach. As I said before, there is nothing we as humans can give them that is more satisfying that killing a chicken or chasing prey or what have you. There has to be a negative consequence for the bad behavior, in addition to teaching the animal an alternate option to chasing/killing (like - being on a "stay" until they learn self control)
I would definitely recommend working with a professional trainer on this, it's not a simple fix.
I would say two huge factors in your scenario is that you were away, and that the goats were babies.
I trained a dog that was menacing people in their foster person's home. I never had a single problem with him and other people, because from day one he was made to understand that I was the leader and in charge of who got to do what. I rehomed him with someone that didn't fully understand the importance of that - she just saw him cuddling with strangers at the park and didn't listen to the rest, I suppose - and did not continue using his training, and he bit someone in her home. He got rehomed with someone with experience with assertive dogs like him, and she never had a single problem with him. It seems like your dogs had a rapport with you that they did not have with their farm sitter. I completely agree with a previous poster's recommendation of boarding them when you're away. Alternatively a very tall chain link enclosure that is sunk into the ground to prevent digging could be used as an place for them to romp with the pet sitter, any kind of k9 alcatraz that lets them get outside time without any access to approaching the goat fencing.
I came home one day to find all eight of our sheep and goats outside their pen with one missing. Turned out the neighbor dog had scared them so bad they trampled down the electric netting and chased one of the sheep deep into the woods (we found her alive and well, thank god). With less aggressive dogs, our horned doe had no qualms about showing them why they ought to leave the goats alone.
And, dogs that have issues with livestock are the most reasonable to rehome. Folks that live in the city likely won't ever have to deal with the fact that their dog is a baby goat killer. As long as you're completely honest about all of the dogs history, you're in the clear. I was able to rehab a confirmed cat assassin (this dog would HUNT cats - silently stalk until she was close enough to grab) after a lot of training. She got to the point where she could be in the same room as a cat and relaxed enough to greet people and get pets rather than fixating on the cat. She lived with me in a house with two cats without incident. BUT, that's because I watched her like a hawk and was training her every moment she was in sight of a cat. She was rehomed with a family without cats and they received training on how to handle her, and they understand the responsibility they have to keep up her training. So it's possible, but it depends on how much work you're willing to put into it.
Also have to put in a plug for pit bulls - they're all individuals. While they are certainly a more high drive dog that generally enjoys a good scrap, my friend's pit was raised on his farm with hogs and chickens. at 4 years old he occasionally enjoys scaring the chickens (he'll run at them until they scatter, and then he walks off, laughing to himself I assume) but has never hurt one. he did attack a pig once, but only that once. He clearly *wants* to go after the pigs but he knows he ought not to, and despite being unsupervised loose on the farm has never gone after even a piglet since his one incidence. Personally I wouldn't take the chance but his owner does and it's worked out for him.
