I really think it's dependent on what the condition is. Mine was regarding a previous surgery and screw placed in my wrist and the screening was performed by a local civilian physician. They basically took some x-rays and checked my range of motion. The physician really didn't explain what the criteria was, he just gave me a piece of paper that said I was granted a medical waiver.
I'm gonna have to check on this with my army MD friends, but this is how it works...
If you have a preexisting condition that could potentially affect your performance in the military, you need to see an authorized military doctor. If after the examination the doc determines that your condition will not impair you in your field, you get the waiver.
Once you're in, and you develop another condition, or your existing condition worsens and starts to affect you, you'd need to get something called a "medical profile" which basically is an excuse slip from the doctor that excludes you from doing certain physical tasks.
For example, a had a friend who had sever asthma (preexisting condition) and (I assume) she was given a waiver. While in the military her condition worsened to the point where she needed surgery. She could not wear a gas mask properly, so she was given a permanent medical profile saying she cannot wear a gas mask. And of course because she can't wear a gas mask, she was exempt from deployment.
The most common profiles I've seen issued are "No running for the APFT" due to bad back or knees, in which case these soldiers do the 2 mile powerwalk instead of the run.