Twin Banter

"Alexa: You're twins. Ian: What? No! Really? Jeremy: Mum should have told us. I feel cheated! Ian: Duped! Jeremy: Hoodwinked! Alexa: They think they're charming. Liana: Rakish. Alexa: Roguish."

- Barbie and the Diamond Castle

A Twin Trope. The tendency of twins to banter with each other. This includes Finishing Each Other's Sentences and feeding each other lines, among other things. A common trait of Single-Minded Twins and Telepathic Twins.

This can also be done by other multiple siblings, or even Those Two Guys as long as they know each other well enough.

Truth in Television, especially with twins of the same gender, as sharing one room and being in the same grade means they spend much more time together than average siblings. For that matter, some sets of twins develop a language specific to themselves that no one else understands.

Anime and Manga

 * Kaoru and Hikaru in Ouran High School Host Club do this regularly.
 * Futakoi.
 * Bakemonogatari's Next Time On sequences are rapid-fire, seemingly random exchanges between Tsukihi and Karin, the main character's younger sisters.

Fan Works

 * Fred and George Weasley do this far more often in Harry Potter fanfiction than they do in the source material. In some fics, they only ever talk this way -- sometimes because of Twin Telepathy or because they're a small hive mind.
 * When he finds himself in the parallel worlds of Dual! Parallel Trouble Adventure, Douglas Sangnoir of Drunkard's Walk is temporarily split into two independent people and gets a kick out of doing this with himself.
 * Ethan and Evan in the Glee fanfic Dalton.

Film

 * In the 1994 film Nell, Jodie Foster plays the surviving member of a pair of twins, whose almost-incomprehensible speech is a mix of heavily-accented (and mispronounced) Biblical vocabulary and a private twin language she created with her sister before she died.

Literature

 * Fred and George Weasley from Harry Potter have a reputation for this among the fans (see Fan Works, above), but don't actually do it all that often.
 * Who could forget Tweedledum and Tweedledee from Through the Looking-Glass, and What Alice Found There?
 * The Corsican Brothers
 * Castor and Pollux Stone in The Rolling Stones by Robert A. Heinlein.
 * Also from Heinlein, Lapis Lazuli "Laz" Long and Lorelei Lee "Lor" Long from Time Enough for Love and subsequent novels.
 * The Belgariad's Beltira and Belkira. "We're sorry, it's - " "Our nature."

Live-Action TV

 * Gem and Gemma from Power Rangers RPM are an extreme case, as it is exceedingly rare for them to individually speak in complete sentences -- usually they alternate words or phrases.
 * The Sklar brothers of Cheap Seats. (It's also a staple of their stand-up routine.)

Music

 * The song "Idioglossary" by Ponyphonic is about a private language created by a pair of siblings and briefly covers some of how it was created. The title of the song is a shout-out to one of the formal terms for such a private language, "idioglossia", as noted below in Real Life.

Recorded and Stand Up Comedy

 * The comedy duo Todd and Adam Stone, twin brothers who performed as Stone and Stone. Sibling Yin-Yang and Incest Subtext ahoy.

Visual Novels

 * Shion and Mion from Higurashi no Naku Koro ni do this occasionally, but sometimes it isn't always for laughs.

Web Comics

 * Marian and Stephanie from Fragile, while far from single-minded, occasionally are "doing some weird twin-chat".
 * Eerie Cuties has sisters of Cessily (and as such snow fairies) Neige and Sorbet.

Web Original

 * The Duumvirate does this without sentence-finishing.

Western Animation

 * Jim and Tim Possible from Kim Possible.
 * Jeremy and Ian from Barbie and the Diamond Castle. They even banter in song.
 * Tomax and Xamot of G.I. Joe.

Real Life

 * Musicians Tegan and Sara are known to do this.
 * The creation of a private language between twins is actually common enough to have a couple formal terms for it in psychological literature: "idioglossia" and "cryptophasia".