Here's the expert answer - this is Sam's full time job, and she's closed the case on hundreds (thousands) of animals:
The reason dogs like Sassafras and Olivia (see her blog here), termed in the industry as "wandering dogs", are so difficult to catch is that they do not have a single central location that they use as a den, but instead just rest wherever they are when they get tired. They develop large circuitous territories that they move around in and are fearful of people so they avoid contact with strangers. They are similar in description to other types of dogs so the owners are frustrated by many false sightings. Sassafras, being a social girl with other dogs, apparently also "packs up" with other dogs, beagles in particular and is not recognized as the missing dog because they are being seen together. Wandering dogs are frequently out for months and sometimes even years because they are extremely resourceful at finding food sources and water sources throughout their territory and move from one to another along their way. It may be days or even weeks between visits to one particular feeding site so getting ahead of them with food is nearly impossible unless you can find a location where you can leave the food and camera for weeks or months. This takes patience, dedication, and a lot of community cooperation which is hard to achieve as most people don't want food left near their home for long periods of time because it attracts vermin as well as stray animals.
These dogs are frequently caught by accident when they get into a yard or building that they cannot get out of before an observant human closes a gate or door to trap them inside. Recently, a lab/bloodhound mix that was out for many months, found himself trapped by a quick thinking person who saw him inside the man's newly constructed home. He was hiding in the upstairs bedroom when the homeowner came in to show friends the new house. Because of the posters in the area he recognized the dog immediately and managed to corner him and keep him there until the volunteers were able to get over there and pick him up.
Other dogs continue to exist on their own until they are lured with food to one single location and then trapped with a humane trap of some kind. The challenge with these dogs is to catch them before they get hit by a car which is always their greatest threat. The key is to keep the dog's picture/posters in front of the community so that they are aware that the dog is in their neighborhood and will report if they see it. This is the only way volunteers can monitor their movements in hope of getting ahead of them with a feeding station.