An important aspect of benchmark hunting and recovery is to take a closeup photograph of the disk. Often a disk is buried under dead plants and leaves, and perhaps some dirt. It is best to clean it up a bit for a picture. Bring a rag, or a pack of pre-moistened wiping cloths or a few paper napkins. Those will clean dirt off of a disk much more efficiently than your fingers. It is not necessary to polish the disk. Just enough cleaning to be able to read all the inscriptions will be sufficient. The most important inscription is of course the ["designation"] stamped on the disk, but it's also important to clean off the rim of the disk where the name of the monumenting agency is written.
Some people bring along some cornstarch or other such materials to whiten the lettering to make it more distinct.
In the case of benchmarks mounted vertically on a building and painted over, it is probably unwise to try to remove the paint since that would detract from the building maintenance efforts. A small mirror and a sunny day might help you to get a lighting effect that would make the lettering stand out better. If the paint is so thick you can't read anything, and the setting is one where your cleaning will not cause problems (e.g., the disk at the base of the town water tower that has been painted 10 times), then the best method is to soak a rag in an organic solvent (e.g. carburator cleaner, nail polish remover, etc) and place it over the disk, and cover with plastic or foil so the solvent does not evaporate. After soaking for a while, the paint should come off by wiping with a rag. Repeated applications may be needed. Never use abrasives, scraper blades, or corrosive cleaners because of the damage they will do to the disk.