When you wear gumshoes, you're pulling on waterproof galoshes over your regular footwear. If you are a gumshoe, on the other hand, you're a private detective, investigating a case.
The "detective" meaning of this word comes from the "galoshes" definition. In fact, during the late 19th century, gumshoes or gums were any type of rubber-soled shoes. Around the turn of the 20th century, to gumshoe meant "to sneak around," and by 1906, gumshoe became a common nickname for plainclothes detectives (or "private eyes"), famous for moving stealthily in their quiet rubber-soled shoes.