I know someone with a dog for her ASD, though she doesn't have Asperger's.
I've thought about one for myself, and am task training my dog for helping me at home.
The thing you need to remember about service dogs is that they need to do tasks - they need to explicitly to do trained things to mitigate your disability, not be there and make you feel better, not do things that are normal dog things to do, and not things that might help someone else but not you.
You also need to realize if you are training a dog yourself that not nearly every dog will be able to take the stress of being a service dog. It's a very high stress job. If they're not specifically being bred and are just random dogs from a shelter for it only about 1/100 make it.
However even if they don't make it they can be helpful at home and in pet friendly places.
Someone who walks out into roads because they don't notice not to having a dog stop whenever they reach a road is a task. A dog that licks your face when you are upset isn't.
Autism service dogs are really useful though
. There's a lot they can do, it just comes down to what each individual needs.