Have a great Pyrenees. He's 3 now, and has free roam of our 8 acres, day and night, chases the bear off really good. When he was younger he would chase chickens, even got a chick in his mouth, but not even in a playful or aggressive manner, just wanted to lick it and love it. (The chick died a few hours later though, probably due to stress, but it could possibly have been squished a bit too hard..)
He is really good with the chickens now, and though he does enjoy to run through them to make them panic and fly, he does not chase them, so I have no issue. He had chased chickens a few times, but he has an electric collar (only way we could have ever gotten him to listen), and got a good big shock a few times when he did that. He now knows that chasing the chickens is bad.
He is the only dog here, and I think that may have contributed to his success in guarding livestock without a fence.
The farm I go to get milk got 3 pyrenees pups and put them in the field with the sheep, but they were more bonded with each other than the livestock, and did a horrible job at protecting anything. They were about a year old when they were lured away by a pack of wolves who killed and ate them.
With pyrs it's really important that the dogs bond to the livestock and know to guard them than being bonded with their fellow dogs or be to preoccupied with the other dog(s) to do their job.