The Barber

Back in some small towns in the old west, The Barber was the only person who had experience handling a very sharp knife (and the association between barbers and surgeons goes back centuries.) The town might not be big enough to have a physician, so in addition to cutting your hair and shaving you, he might also do some "minor" surgeries, or any major ones where you either didn't have enough time to get to a real doctor, or would probably die from the rigors of the trip. Sometimes was also the mortician since he had to shave the corpses, too. This type of Barber might show up in a "realist fantasy" that has done the research.

In fictional Westerns however Barbers are used primarily as sources of gossip, and a local gathering place outside of the Saloon. Expect the villain to show up and demand a shave before he confronts The Hero, thus allowing the Barber to send a warning to The Sheriff. Occasionally can be reversed.

By ancient tradition, the barber tends to be extremely talkative, often taking advantage of having a captive audience who can't reply because they've either got a hot towel on their face or a cutthroat razor at their throat.

Anime and Manga

Part Three of JoJo's Bizarre Adventure has a barber named Khan, who's a largely normal person... until he accidentally unsheathes Anubis.

Comic Books

Jolyon Wagg from Tintin has a barber uncle. That would explain where he got it from...

Film

Billy Bob Thornton's character in the Coen brothers' The Man Who Wasn't There is a subversion of the stereotypically friendly, chatty barber: he's a simple, dull fellow who leads a life of quiet desperation that abruptly gets loud.
The Gunfighter uses the barbershop first as a place for The Gunfighter Wannabe to learn that The Gunslinger has come into town. The barber then warns The Sheriff, allowing him to intervene.
Stan Lee's obligatory cameo in Thor: Ragnarok is a nutty barber in the Grandmaster's gladiator arena.

Literature

"The Tale Of The Barber" in the Arabian Nights, in which a barber tells several stories about his family and chatters endlessly while swearing up and down that he is a man of few but wise words, despite his penchant for chatter and meddling only causing trouble wherever he goes. The end result (depending on the translation) is hilarious.
Little Benjamin in Henry Fielding's Tom Jones.

Live-Action TV

While more of the "chatty captor" than the "surgeon on the side" type, The Andy Griffith Show had Floyd the Barber, referenced in the opening song for Freakazoid!!
Archie Campbell's barber on Hee Haw was a similar character.
The townsfolk of Colorado Springs, Colorado relied on their barber for all their medical needs until Dr. Quinn, Medicine Woman came to town—and for quite a while afterwards too, since they didn't hold no truck with wimmin doin doctorin.
In The Armando Iannucci Shows, Armando's barber is definitely more of a Chatty Barber than a Chatty Hairdresser, nonetheless his chattiness is excessive.
Starting with the Candy Store series, some later Gospel Bill videos (a Christian Western series) introduce a barber called "Lefty". True to form, he doubles as an undertaker.

Newspaper Comics

Gunther the barber in Curtis is basically a male Chatty Hairdresser, who always gets the protagonist's name wrong.
On Blondie, Dagwood's barber has a weird sense of humor, as shown in this Halloween strip.
Charlie Brown's dad from Peanuts is a barber, much like Charles Schulz's dad was.

Radio

An excellent example of the chatterbox style of barber appears in an episode of The Six Shooter Radio Drama delivering an info dump on the goings-on in the town as the hero Britt Ponsett (James Stewart) struggles to get a word in edgeways.

Theatre

Man of La Mancha, which actually features a journeyman barber who still loves the sound of his own voice.
Figaro, The Barber of Seville.
Sweeney Todd: The Demon Barber of Fleet Street.

Web Original

Played with in the Transformation Story Saga aptly titled The Barber; the main character not only takes care of his customers' hair, but also of their physical wellbeing... by making them become somebody else (always a very attractive, gay man).

Western Animation

Courage the Cowardly Dog has Cousin Fred, a parody of Sweeney Todd that while equally creepy and psychotic merely has an obsession with cutting hair (in excess) rather than throats.
Doctor Barber from The Marvelous Misadventures of Flapjack, who also dabbles in Weird Science.
- He's also a Candyologist.
In Minesota Cuke and the Search For Sampson's Hairbrush, when Cuke goes to Seville, Italy looking for the titular brush, guess who he hits up for information?
On King of the Hill, this is Bill's job at the military base, which he takes very seriously. Although he is rather proud of the rare times he gets to wear his uniform and do more important soldier-work.
Bugs Bunny acts as a barber for Elmer Fudd while synchronized with opera music in the Looney Tunes short The Rabbit of Seville. He's not at all talkative, though; he's too busy singing.