I don't have kids, but I've gleaned some basic etiquette for when co-workers bring tots to the office. Turn to introduce yourself. Smile. Look 'em in the eye. Extend a hand. Be kind, be gentle, offer a sincere compliment. Address them in a bright tone of voice. Learn their name the first time you hear it. Don't give them candy without an adult's permission. And if their parents say they're cool with it, go ahead and toss 'em a ball.
You already see where I'm going with this, I'm sure. People who exalt their dogs to childlike status become annoying in a hurry, and my saying this isn't meant to equate dogs with toddlers, but if you're wondering how to deal when a co-worker introduces a dog to the workplace, just imagine what you'd do when meeting a child who's just big enough to walk around and who is excited by this new space full of new people. Depends on the dog, of course — some truly are idiots — but you can generally count on a dog to exhibit the intellectual capacity of a 1-year-old child. In a pinch, they make a passable shorthand.
Dog owners (or dog-sitters, as I was when the mini-monster in the photo above came to work with me one week) are going to be running interspecies diplomacy. When I cleared the path for this dog to be the first allowed into an office, I started by reaching out to the office manager and explaining that he's quiet, housebroken, hypoallergenic, and chill. Fortunately he was truly all of these things, content to explore people's feet and chew on rawhide treats and get any scritching that folks offered him, all without making a ruckus or a mess.
If someone smuggles a dog into the workplace who's rambunctious or loud or pisses under your desk, or who snarls, or who chews on your purse, then you should pull a manager aside — again, with a couth you'd bring if discussing someone's 3-year-old — and let them know that you're having trouble getting your work done with the dog around. No judgment on the dog! Dogs can be fantastic at being dogs and yet not great at helping work get done. Framing your grievance an as issue of productivity or concentration is an office language withan unmistakable subtext: Work trumps (literal) creature comforts, and if push comes to shove, the person trying to get work done around here will have the stronger claim.
No one wants it to come to that. And it shouldn't, if the dog's master and his new friends are showing mutual respect. I once worked in a company where on occasion the owner's surly dachshund would blithely shit in the hallways, and there wasn't much to be done about that, because hashtag like a boss. A fireable employee doing his best to integrate a morale-boosting quasi-mascot into your daily life truly wants everyone to get along. If someone brings a dog in, ask its name, ask whether you can pet it, do not presume to be allowed to pick it up or feed it, and let the owner know your general level of comfort with dogs, especially if that level is: You love dogs or have animals of your own. Quietly the dog person will be looking for allies. If you want to see the dog return, say so. If you don't, say that to a person who makes officewide decisions. In any case, to be safe, imagine you're talking about that person's kid. You just never know how people see that critter they're lugging around.