How do I teach or train my own dog to not be aggressive with client guest dogs? My dog, I feel, feels threaten and does not understand. My dog will growl and shows her teeth and acts like she wants to attack but doesn't?

First thing's first, BE CAREFUL! It's possible your dog is not ready to be a dog sitting host with you and needs some additional warm up time, training, practice, etc. With that being said... Where do you generally meet client dogs? I always meet new guests outside on slightly more neutral ground than inside the house. I would also recommend thoroughly exercising your dog before meeting new dogs. Make sure you are calm! If you are nervous, worried, anxious, etc, your dog may feel the need to protect you from what it perceives as a threat. Try walking by the new dog instead of facing him/her. Try going for a brief walk with the new dog (and their owner if they have time). I usually try to limit the direct face-to-face contact between dogs when there is any kind of tension between them and do a side-by-side walk or pass-by instead. Try to relax the leash so they're not physically uncomfortable from stress on their neck, which can exacerbate the tension between the dogs.

Do your research! Start watching dog behavior and training videos so you can determine the possible cause of your dog's discomfort (e.g. anxiety, discomfort with the other dog's behavior, protectiveness, territorial behavior, etc).

My dogs would behave similar at first but with each meet and greet their behavior improves each time. I have learned to have new dog off leash in the my yard, comfortable. They I let my younger dog out first to get over her extreme excitement. I then let the older dog out who finds everyone happy and playing. When they are all together at once it is overwhelming for all dogs. You could try meet and greet at the owner's home with your dog. Have you tried that instead?