I'd say there is no right answer to how to train your dogs to be around chickens. With our labs, all we've had to do is say "no" twice to them. That's it. They know to leave them alone. Sure, they follow them around, but just to eat their poop. I trust each of our lab alone out with the chickens. The dachshund has been more of a project though. She has such a strong hunting instinct, that we haven't been able to get her to stop hounding them. Now, after a year and a half of chickens it seems that she might have mellowed out to them though. Karin had her loose in the yard with the chickens out, and she didn't care about them. I wouldn't trust her alone with them though, the wrong kind of sprint from the chickens might trigger her instincts at any moment. And she kills in an instant, only needs a second or two to dispatch a chicken.
Dachshunds and killing.... This was in San Antonio, Texas, early '60s , we went to the mall in the Spring, near Easter, and there was a vendor selling chicks and ducklings. One of the ducklings was very pitiful looking and my sister, about 7 or 8, asked about it. The vendor said it was free if we bought another one. My parents, both raised on farms, knew the weakling wasn't going to make it but agreed to buy one so my sister could "save" the weakling. The weakling did die but the other duck was a male Pekin, Charlie, that we spoiled and loved. He got along with our Beagle and my dad built him a "duck house". We spent hours collecting "June bugs" to bring back and feed him. He loved watermelon and would walk up and take a bite out of your slice if you weren't careful with it. We had next-door neighbors that had a long-haired Dachshund, Buckwheat. He would run along the chainlink fence barking at Charlie and we were always careful to never leave the gates to the backyard open....until one day when my neighbors, they had a girl my age and we were fast friends, came back from vacation. We ran out of the yard, leaving the gate open, and they had opened their garage which had a "doggie door" for Buckwheat. He never looked at his family, he ran straight toward our open gate. We were right behind him...I mean right behind him...he killed Charlie in the instant he reached him. We were all stunned at how quick it was. My Dad buried him in the backyard. It was the first time I saw my Dad cry, I was 5 or 6. You gotta love a man with a heart for animals.