You've actually got a lot more power than you realize. First, you control each pet's grooming experience through the way you conduct yourself. The animal will probably sense your calm, competent demeanor and confident grooming skills.
The right attitude puts you in command. On the other hand, animals will also pick up on your uncertainty and lack of confidence, and might even use that opportunity to their advantage. You get to choose the outcome.
You also have the power to determine the finished result. Hopefully, you try to make each pet look their best regardless of how cooperative (or uncooperative) they are. We also hope you don't take your dislike for a pet owner out on their dog, and give them the weirdest style possible.
Remember, you decide whether you'll give that standard poodle a gorgeous show cut, or make him look like a strange garden arrangement. That does sound strangely appealing, though. Next time a poodle owner says, "Use your own judgment," that's what she's gonna get.