I see that you are a studio owner and personal trainer yourself, and I assume you want to know what traits potential clients are looking for in a personal trainer.
Here is what I tell people who ask me what they should look for in a personal trainer:
Ask about qualifications and experience and see if you can talk to a current or previous client of the trainer who may be similar in requirements to your own. For example, if you have back problems, make sure the trainer actually has practical experience rather than just theoretical one.
If the trainer is willing to train you right off the bat without any assessment, run the other way! Those famous “free personal training sessions” can come at a high price. Make sure the trainer listens to you and takes notes. You should have her full atttention. A trainer should be able to explain why he picked each exercise and modality.