Warning: SPOILERS for Captain America: Sentinel of Liberty #8
Marvel's that the term "Secret Avengers" is now common knowledge.
Initially, Captain America was not the leader of the Avengers. That honor went to Iron Man, but when Captain America was revived from the ice in the 60s, he eventually took over team leadership; his skills in battle, tactical knowhow, and his ability to keep a group of volatile teammates working together were all indefensible. Whenever Steve Rogers is unable to serve as team leader (either thanks to other obligations or even being dead), others step up to the plate: Iron Man and Hawkeye have assumed the mantle, but Captain America is usually the man on the front lines.
In Captain America: Sentinel of Liberty #8, written by Jackson Lanzing & Collin Kelly with art by Carmen Carnero, Captain America awakes in a location in Kansas after a plan to defeat AIM from taking over New York goes horribly wrong. He dashes to a gas station to use the attendant's phone, and unfortunately, the Avengers are unable to be mobilized (ever since the aftermath of the Devil's Reign event). "Secret Avengers?" asks the attendant, and Captain America muses that this is a good idea. He ends up calling the Captains Network, a group of self-appointed Captain Americas spread out across the country.
Everybody Knows About The Secret Avengers
The Secret Avengers were created after the events of Dark Reign, and Captain America himself was tasked with assembling the team. Unfortunately, the main tenants of the Secret Avengers ("Run the mission. Don't get seen") fell by the wayside as the team fought larger and larger threats. The formation of the team was initially seen as a way to keep the normal Avengers far from morally-gray work, but both teams have had their share of nebulous actions (Civil War, Fear Itself and the Avengers vs X-Men event, to name a few stories).
It must be noted that secrets in the Marvel Universe, especially when it comes to teams, rarely stay secret for long. The Illuminati became public knowledge after the events of World War Hulk, and the famous X-Men team in 1975 were actually the second team formed; Professor Xavier's first team was a failure and kept a secret for years. Captain America's team of Secret Avengers fail to live up to their name, but they are, at the very least, respected within the country.