Introducing a second dog into a household is not always easy. When the new dog is an Great Pyrenees, a few extra precautions are in order.
If this is your first Pyr, you have - hopefully - researched the breed, talked to other owners, and are prepared physically to have a large dog in residence. You have a fenced yard, a kennel or crate to confine the Pyr when you can't watch her, you have purchased a chain collar of exactly the right size, a six-foot leather leash and a 12-inch "Traffic Handle". You have purchased a large food dish and laid in a supply of appropriate dog food. You understand that a Pyr is a territory-protection dog; that they can become "industrial-strength" barkers when bored as well as prodigious diggers. Despite their often "teddy-bear" appearance, they can be fierce, deadly fighters when defending their territory. It is up to you to define that territory and to introduce your Pyr to its new "flock".
If you already have a resident Pyr, extreme caution in introducing two adults is in order. If the new Pyr is a pup, the introductions are generally much easier. When introducing adults, opposite-sex introductions are generally the easiest, same-sex introductions more difficult. Unless you have many years of experience with this breed, and have the assistance and advice of someone very familiar with the background of both dogs DO NOT try to introduce another male into the household with a resident male Pyr.
On homecoming day, take your resident dog(s) for a leashed walk in a neutral area - preferably a park or some place they are used to seeing non-threatening dogs. Have a third party approach with the new Pyr and, keeping all dogs on leash, allow introductions to take place naturally. Discipline any dog that acts aggressively. Once introductions are made, take the new Pyr's leash yourself and continue to spend some time with all the dogs. Do not sit on the ground with them - you need to be able to get out of the way if someone starts a fight and being taller than the dogs emphasizes your "alpha" status to them. Walk toward home, talking naturally to all the dogs, but not showing preference to any of them.
When you get to the house, keep all dogs on the leash for the first half-hour. The new Pyr will want to walk around the perimeter of the yard, marking as she goes. Give her a drink of water, then put her in her kennel or crate. Spend some time with your resident dogs, put their leashes away, offer them water (but no treats yet) and then let them get acquainted with the confined Pyr. Discipline any signs of aggression on anyone's part. Once all have lost interest in each other, give them each a treat - with the new Pyr still confined - and go on about your business for a while.
Later in the day, leash the new Pyr, allow her to relieve herself, and bring her into the house. For the first several weeks, bring her in the house only on a leash. This will facilitate house-training as well as give you control in case of a problem. If there is a resident Pyr, leash it only if it shows aggression toward the new dog.
Dogs have a natural understanding of pack-order. You as human are - or should be - the alpha. But within the pack itself, there is a subordinate order that, when understood, will help you better modify their behavior. If you have only one resident dog, then it obviously considers itself higher in status than the newcomer. Don't allow the newcomer to challenge that assumption. If the resident is a male, and the newcomer a female, the order will work itself out in time, and the female will become higher ranking than the male in most cases. If both are female, watch for signs of the newcomer trying to challenge the status quo. If the newcomer is a pup, don't worry about growls and snaps in the air by the adults; this is normal puppy-training behavior and pups are very rarely injured by adults.
If you have only one other resident dog, you will probably be able to allow them to play freely together within the first week. Be certain that no toys, bones or food are in the play area at first. If you have several resident dogs, they are already a "pack" and if one member of that pack takes a dislike to the newcomer, they might all cooperate in a fight and someone will get hurt. Take a much longer time in introducing the newcomer to a resident pack; keep her on the leash at all times until you are absolutely certain that everyone has accepted her. This may take several weeks.
When feeding and giving treats, respect the pack's established order. The highest-ranking dog is fed first, and so on down the line with the newcomer getting hers last. If you free-feed (have food available all the time) it may help to have more food sources than dogs, so no one has to "protect" their dish. If toys are available, always have more toys than dogs to lessen the chance of hoarding and fighting. If you walk all of the dogs at once, put the new Pyr on the traffic handle until you are certain you have complete control of her. An unexpected bolt from the end of a six-foot leash can have serious consequences for both of you.
Taking the time to introduce a new Pyr to the resident dogs will pay off in fewer squabbles and quicker integration into the household.