I think you can answer this question by asking yourself whether you would want this outcome if your dog went astray. Say you were on vacation for two weeks and your dog got away from the dog walker or neighborhood kid you hired to care for your dog while you were gone. Someone found the dog, but because they didn’t get a response to their found-dog posts or the report at the shelter, they assumed the dog had been “dumped” and gave the dog to their brother-in-law (or someone else). That person, your dog’s new owner, might never be aware that “your” dog is very much loved and missed and wanted.
If you fall in love with the dog, and are tempted to keep him yourself, just keep in mind that there can be any number of reasons why a person may be unable to search for their beloved dog, or see and respond to “found dog” ads. There was just an article in the Washington Post about a man who was in the hospital, with a friend taking care of his little dog. His condition worsened, and his doctors put him into a medically induced coma for over three weeks. Sometime after he was reawakened, his friend came to tell him the sad news that his dog had run away weeks before. The dog had been picked up and taken to a shelter, where, after being unclaimed for weeks, he was put up for adoption – and was adopted by a person who fostered him for the shelter.
Fortunately, an intermediary who was familiar with the shelter connected the adopter and the dog’s original owner, and acted as an intermediary to convince the new owner that the little dog really did belong with his original owner, who lost the dog through no fault of his own. But the media is full of stories like this with more contentious endings, where the new owners won’t give up their newly adopted dog, and the shelter is unable to legally intervene.
It’s best if you can host the dog yourself while continuing to share occasional found-dog posts, at least for a couple months, before determining that no one is out there still looking for their dog. Ask your local shelter staff what the law in your state requires before you can consider a “found dog” your own, and when sufficient time has passed, get the dog licensed in your name and get microchipped – and make sure you register it as soon as possible.