Averted more often than not in the TV shows. While the operators all perform multiple characters, they must operate each Muppet as well as do the voice, making it virtually impossible for one performer to handle two characters at once. So we seldom, for instance, see Fozzie and Miss Piggy (both originally performed by Frank Oz) interact. When they do interact, the standard method is to pre-record the character that is easier to handle. This is why, for example, it was Floyd Pepper (performed by Jerry Nelson) who represented the Electric Mayhem band in backstage discussions with Kermit the Frog (performed by Jim Henson), even though Dr. Teeth (also performed by Henson) was actually the band's leader.
The feature films, on the other hand, enabled more of this to occur. The Muppet Movie, for instance, features a duet with Kermit the Frog and Rowlf the Dog, both voiced at the time by Jim Henson.
Big Bird and Oscar are both voiced by Caroll Spinney; in scenes with the two of them together Spinney would pre-record Oscar's dialog and someone else would operate him, since he's easier to handle than full-body Muppet Big Bird. This has changed a little as the performers have grown older, retired and/or had health issues. Carroll Spinney no longer operates Big Bird or Oscar; his understudies Matt Vogel (Big Bird) and Eric Jacobson (Oscar) now puppeteer the characters. Jerry Nelson, on the other hand, handled only the voices of his characters and not the puppetry from his retirement in 2004 up until his death in 2012.
One impressive bit by ventriloquist Jeff Dunham involves him getting in a rapidfire three-way argument with two of his characters. In another routine, the same two characters start having a conversation in presumably fluent Spanish, and Jeff remarks that he feels left out because "I don't speak Spanish!" No wonder one of his shows is called Arguing with Myself.
Technically, this trope applies to all ventriloquists, as the traditional format is of them having a conversation with the dummy. The fact that Dunham can do this with multiple dolls at the same time just shows what a master of the technique he is.
In one of his early skits, his puppet, Peanut, has his own puppet. It involved Dunham, Peanut, Peanut's puppet Mini-Jeff, Jose Jalapeño (On a Stick), and the worm at the bottom of a bottle of wine. You can see it for yourself here.
That skit seems to have returned in Dunham's latest installment, Controlled Chaos. Only it's a conversation/argument among Jeff, Peanut, Mini-Jeff and Mini-Peanut (a hand puppet of Peanut on Jeff's other hand).
Peanut also does some epic lampshading of ventriloquism in Spark of Insanity as he tells Jeff, "We cannot talk at the same time!"
Oobi: Stephanie D'Abruzzo plays both Uma (Oobi's sister) and Inka (Oobi's piano teacher), so the crew had to find creative ways to fit them both in the same scenes without having her actually play both characters. For example, one episode has Inka call her date, Grampu, from off screen outside while Uma and Grampu are inside. Another has Oobi and Uma find a Valentine's Day card that Inka delivered to Grampu, but Inka is never seen. Possibly the most obvious example is in the "Recital!" episode, which heavily features Inka. Uma is said to be "visiting Aunt Oota's" for the entirety of the story.
British kids' show Rainbow features George and Zippy, a classic Odd Couple who are continually arguing with each other, interrupting and talking over each other — despite being both products of the same actor who is not only Talking to Himself but doing it live.
The 1960s British marionette science fiction show Space Patrol, a.k.a. Planet Patrol in the United States, has a very small voice cast who often play multiple roles, and according to one interview they would simply switch voices while recording their lines, rather than relying on editing.