You've actually got a lot more power than you realize. First, you control each dog's grooming experience through the way you conduct yourself. The dog will sense your calm, competent demeanor and confident grooming skills, and will correctly get the idea that you're in charge. On the other hand, the dog will also pick up on your uncertainty and lack of confidence, and might even use that opportunity to his advantage. You get to choose the outcome.
You also have the power to determine the finished result. Hopefully, you try to make each dog look his best regardless of how cooperative (or uncooperative) he is. Hopefully, you also don't take your dislike for a pet owner out on the dog, and chop him up so he is almost unrecognizable. Remember, you decide whether you'll give that standard poodle a gorgeous show cut, or whether you'll turn him into a dreadlocked Rastafarian dog who just hopped off the plane from Jamaica. That Rasta poodle cut sounds strangely appealing, though. Next time a poodle owner says “Use your own judgment,” that's what she's gonna get.