I wonder whether actually many of these dogs aren't being stolen but that if a dog disapppears out of a garden then suggesting that it could have been stolen is far more likely to get a positive share than if the dog had managed to get out and wander off. I'm not saying that dogs aren't stolen, but given that many dogs these days are newtered there would be little market for a newtered four/six year old dog, for instance...
Last year there was a very high profile case of a guide dog going missing in Scotland. I can't link because on my phone, but basically the owner was free-running her dog and it went missing. For weeks, months even there were posters all over fb, with links to printable posters for people to share/put up all over the country. The story being shared was that it was believed this dog could have been stolen and could be anywhere in the country by now.
The reality is that this dog was running in a fairly open area near farmland, it is in fact far more likely that she ran on to a farm and was possibly shot by a farmer, although no farmer would have wanted to come out and admit that he'd shot a guide dog given the high profile nature of her disappearance, even though he wouldn't have been in the wrong for shooting a dog runing on to his farm and possibly scaring his livestock even if not agressively...
But once her disappearance was shared on fb guide dogs had to take it over and every possibility had to be explored. But they also will have had to be careful to avoid any criticism possibly directed at the owner/or the organisation.
After all imagine someone coming on here posting "my dog has wandered out of my garden," although some would be sympathetic there would be the inevitable calls of "why was your garden not secure? if you'd secured your garden the dog would still be here, you irresponsible dog owner..." and if someone posted that they'd been running their dog close to open farmland and she'd disappeared they would be called irresponsible for running a dog close to areas where livestock may be grazing and that the dog might have been shot and it would be all their fault.
Fwiw in this instance I don't think anyone did anything wrong, and as a guide dog owner I can't imagine what the poor owner here went through.