I had no idea these schemes existed but can fully vouch for the benefits a dog brings. My thoughts on pet dogs for those who don't qualify for the schemes mentioned above:-
Breed selection is critical. Labradors and retrievers or 1st generation crosses are the "standard" assistance breeds so would be the first breed I'd look at - only moving onto something different if for some reason these breeds were unsuitable for my situation. For those who can devote a lot of time to excercising the dog a spaniel is nice dog (very affectionate) for kids generally and was on my breed shortlist. Sight hounds, (cuddle cravers)are generally gentle..
My advice to other Mums who want a dog to help their child "break the ice" when socialising in the park, on the school run etc is not to get a staff, or any of the breeds commonly demonised in the media. You want the dog to help break the ice, not put up more barriers (I do know this is unfair to staffs etc btw). I wouldn't rec a terrier for anyone whose child has poor empathy as they don't often have much patience. I'd also stay away from the traditional "guarding breeds" just in case the dog ever misinterprets the child's distress in a social situation, (just my own view having a child that suffers severe anxiety - I want the dog to calm & comfort, not " protect" iykwim).
Don't feel you HAVE to get a puppy, I knew wanted a young adult for a variety of reasons, some familes might prefer an old codger. Lots do a very good dog fostering scheme, which is a nice option for those who don't want the full responsibiliity of life time ownership or feel that an older dog would better suit their lifestyle & needs. Generally breed rescues do a much more personalised service than the big rescues, so are worth listening to when making your individual selection. They also "get" waiting till the perfect fit comes along, rather than taking the first fit female between X&Y years old - this was really important to me, and I really feel it paid off.
We eventually got a young whippet via lurcher link and she's perfect for DS, after an initial 12 months getting used to our funny ways. Do be prepared for the bond between child and dog to take a good 12 months to fully develop - this isn't an "instant fix". Ensure both child and dog have an exit strategy at all times while they adjust to each other, and NEVER leave them alone together.
She relaxes me too, not just DS. You have to genuinely like dogs and be prepared for insurance, walks etc. Training her to stay silent on the floor in cars/buses etc and to sit quietly by my feet means she gets into lots of places she strictly shouldn't from non-dog loving relatives homes to cafes ; ) The fact she's clean and doesn't chew or jump on furniture makes it easy to get friends to mind her when we go on holiday. It makes a huge difference if you train the dog with non-dog lovers needs in mind as to how restrictive you eventually find owning a dog. You'll notice official assistance & guide dogs behavior in public is impeccable - aim for the same. I use the same commands for child and dog for crossing the road, coming when called etc, which has raised odd looks but it keeps DS under control lol!