Wncsnake

Thanks for the question, as it is one I love to answer. The interesting thing about PTSD is it presents differently with every veteran, and to varying degrees. For me, it is being anxious in social situations, avoidance, hypervigiliance, depression and nightmares. My dog knows intuitively when something is up, and sometimes I will space out and she brings me back by licking my hands. When I am really stressed or having a panic attack I can sit down, make a few kissing noises, and my dog knows that means I need her help. She puts her paws on my shoulders, and licks my face. I can also just have her sit between my legs and I can hug her/kiss on her, and her smell and texture help me block everything else out.
Shane has also noticed that having a larger dog helps with PTSD, as people notice and will tend to keep a wider berth, so you are in your own little bubble.
And, as a side effect, Kiowa helps me with my social anxiety since we have been doing this, as I am in groups and talk to people, with them crowding around me and asking questions. I can now do this for a relatively long time, so she has helped increase me comfort zone and made my fight or flight response lessened in that circumstance.

After reading back through your question, I realized I did not answer your first question. I am not sure what PTSD looks like to a dog, but I picture it as this: A dog, unless trained by humans, loves everyone regardless of race, gender, whatever. When you bring the dog into your home, they form a bond with you and your family, and providing it is a happy home that is not stressful for a dog, they still love everyone. When you begin to train the dog, you become more than an object of affection, you become this beacon in their world, because when they do the things you ask, they get rewards (we believe in ONLY positive reinforcement, never be rough with your dogs no matter how frustrated you can get. It betrays their trust and violates their contract with you) so the dog now focuses on pleasing you, to get rewards. When the dog is asked to become a service dog for PTSD, they are trained to recognize the symptoms and bring you back to the present and center you. You are their beacon, and when your light starts to fade, they have a little torch in their mouth to rekindle you. After a while, it becomes less about the treats and pets and more about taking care of you, because that is what they do. You are their life.
This isn't based out of any science, just observations, and my love for my dogs.

REally just talk to us, we try to keep it as easy as possible for the veteran, and we will walk you through the entire process to make it as successful as possible

You can! Dogs are only lazy because you haven't found what really motivates them. My dog did well enough with treats in training, but then I started to use a tennis ball, and it is amazing how fast she progressed. That little fuzzy ball is the best treat she can ever get. If you want any training advice there is a lot of resources online, you can usually get a dog to sit, lay down, and wave/shake in an afternoon.

I think the coolest thing I have seen is the indirect mobility training Draven has received. Shane has balance issues, and if you look at the leash Shane carries, it goes around his chest, so when he starts to teeter it pulls against Draven, and since he is a 100+ pound dog, Shane doesn't fall. Their partnership is so cool to watch.