It varies from superhero to superhero. This answer will contain spoilers for almost everything in the Marvel Cinematic Universe up until the end of Doctor Strange, and is accurate (as far as I know) from the end of that film.
Superheroes whose public persona is well-known
- Tony Stark revealed to the world that he is Iron Man in a press conference at the end of Iron Man.
- James 'Rhodey' Rhodes never attempted to hide his secret, being well-known as an employee of the US Air Force both before and after he took on the War Machine armour.
- Bruce Banner's work as an Avenger means that he doesn't have a secret identity, proven by the fact that there is a discussion about calling for his arrest after the Hulkbuster incident in Johannesburg in Avengers: Age of Ultron.
- Natasha Romanoff had all of her covers blown when she leaked SHIELD files at the end of Captain America: The Winter Soldier. Since then, she's worked publicly as an Avenger, so we know she doesn't have a secret identity.
- Thor doesn't even use a codename, and we see in Thor: The Dark World that bystanders are willing to put themselves in harm's way in order to get a photo of him.
- There is an entire museum exhibit dedicated to Steve Rogers and his history in World War 2 as Captain America.
- A news report in Captain America: Civil War confirms that the public at large know that James Buchanan 'Bucky' Barnes is also The Winter Soldier, an infamous Hydra assassin.
- Sam Wilson's time as an Avenger means that he does not have a secret identity. His name is explicitly a part of the Sokovia Accords.
- The same is true of Wanda Maximoff, who is well-known thanks to her time with the Avengers and the disaster in Lagos.
- The Vision doesn't exist as anything other than The Vision.
- Frank Castle is well-known to be The Punisher thanks to a very high-profile trial in the second season of Daredevil.
- T'Challa does not hide the fact that he is the Black Panther from anyone. It's a title passed down from warrior to warrior.
- Luke Cage and his powers became public knowledge when he decides to use his powers to help out the people of Harlem in the first season of Luke Cage.
Superheroes with a secret identity
- Matthew Murdock is a blind lawyer by day and an acrobatic crime fighter called Daredevil by night, who guards his secret identity incredibly carefully. Only a handful of people know his secret by the end of the second season of Daredevil.
- Peter Parker guards his secret identity closely in order to protect those he loves from retribution from those that hate Spider-Man. It would seem that the only person who knows his secret identity is Tony Stark.
- Robbie Reyes is a mechanic who is also possessed by a spirit of vengeance, and as the Ghost Rider kills people on behalf of those who have been wronged. Very few people know that Robbie is the Ghost Rider, something he is actively trying to hide from his younger brother.
Superheroes that used to maintain a secret identity
- Hank Pym used to work as Ant-Man for SHIELD, but quit following the death of his wife in 1989.
- Jessica Jones gave up her attempts at being a superhero after a traumatic encounter with a man who can control minds left her with severe PTSD. Her time as a superhero is only known by a few people, but she doesn't really hide her powers either.
Superheroes that are not public knowledge
- Clint Barton managed to keep his entire family a secret, and retired from the Avengers before the Sokovia Accords could come into effect, so it's likely that people at large don't know that he is Hawkeye. However, his arrest at the end of Captain America: Civil War means that his real identity may be known to the public; it depends how secretive The Raft and its contents are.
- Daisy Johnson is setting up her own team of superheroes under the supervision of Director Coulson in Agents of SHIELD. In the season two finale, Director Coulson indicates that this team is to operate in secret. Daisy may not hide who she is, but the team isn't known to the public either.
- Scott Lang does not attempt to hide who he was from people in Ant-Man. However, his arrest and imprisonment in Captain America: Civil War means that his real identity may be now public knowledge; as with Clint Barton, it depends how secretive The Raft and its contents are.
- As of the end of Ant-Man, Hope van Dyne appears to be taking on the mantle of The Wasp. We do not yet know if she intends to keep her identity a secret.
- Lincoln Campbell, Joey Gutierrez and Elena Rodriguez are members of SHIELD's Secret Warriors. However, they have also been publicly outed as wanted by authorities. It is likely that they do have secret identities, as they appear to have normal lives outside of SHIELD.
- Elektra Natchios has been working to fight The Hand for some time, and although The Hand (and some of the people who fight The Hand) know who she is, they all operate in the shadows; meaning that the general public have no idea she even exists.
- Doctor Stephen Strange works in secret to protect the Earth from inter-dimensional and mystical threats. It is likely that the general public have no idea that he exists.
- The same is true of The Ancient One, Karl Mordo and Wong.
I haven't included the members of the Guardians of the Galaxy in this answer as the people of Earth aren't even aware that they exist.
