David Crosby and Graham Nash
Willie Nelson and Merle Haggard on “The Ballad of Pancho and Lefty” was the first song that came to mind.
Travis Tritt and Marty Stuart dueted on “The Whiskey Ain’t Workin’ Anymore.” (the song starts around 1:05). It’s not a song that needs two voices, but these two have so much fun with it, I think it would lose something with only one of them on vocals.
Zac Brown Band with Alan Jackson on “As She’s Walking Away.”. It’s a well-crafted song, and I think Zac Brown was one of the better things to happen to Country music in a while, but when Jackson drops in with, “Son, I missed my chance,” he brings some real regret to the lyric. His voice has aged nicely, and it feels like a bit of torch-passing from one generation to the next.
“Au fond du temple saint” from the Pearl Fishers, Bizet
Crane
Sure ain’t a duet. (yeah I know you were suspended about 13 years ago - tough)
Now this is a duet.
I’m surprised nobody mentioned:
Ebony and Ivory - Paul McCartney and Stevie Wonder
Fragile - Sting and Stevie Wonder
To All the Girls I Loved Before - Willie Nelson and Julio Iglesias
I’d Really Love to See You Tonight - England Dan and John Ford Coley
That’s just off the top of my head.
OP is probably referring to Dialogue.
As mentioned upthread, a lot of times the singers aren’t expressing different points of view but rather, they’re singing the same thing in harmony. Peter and Gordon on “World Without Love” or Chad and Jeremy on “A Summer Song” and a whole lot of Everly Bros or Simon and Garfunkel. A famous one by S&G, in which the singers aren’t only harmonizing:
A favorite of mine—sharing the lead, but also harmonizing:
Has that god-awful “More Than Words” by Exteme been mentioned yet?
A number of songs where Simon and Garfunkel either harmonized or doubled for the entire length of a number, (“Scarborough Fair” and “Sounds of Silence” to name a few) would qualify to this listener as duets.
Johnny Cash did a duet with Nick Cave on “I’m So Lonesome I Could Cry” on Cash’s “American IV” album.
Warren Zevon with Bruce Springsteen on “Disorder In The House” on Zevon’s final album “The Wind”.
I win this thread.
My username comes from this song.
This song does not have an official video, but was on the same album.
Weird Al Yankovic did a style parody of the “More Than Words” video. One of his other musicians also sings lead on some tunes, and that includes the “Spongebob Squarepants” theme.
Bowie and Crosby doing “Little Drummer Boy”. 
Alice in Chains, with Jerry Cantrell (lead guitar and backup vocals) and the late Layne Staley (former lead vocals) and current singer William Duvall are known for their harmonies, though I don’t know if that exactly fits the bill.
Oh, that is a great song! Perfect blend of baritone and falsetto.
Willie and Ray Charles on “Seven Spanish Angels” too.
I think I’d also count “Pancho and Lefty” between Willie and Merle Haggard too.
Speaking of “Sounds of Silence”, there’s the duet with Myles Kennedy (Alter Bridge) joining David Draiman with Disturbed:
And Jimmy sometimes drops in when the Zac Brown Band is playing.
They did a great version of “Knee Deep” together. They alternate vocals on the verses, harmonize on the chorus, and trade off each line of the bridge.
I’m not a country fan, but this makes me smile… and feel that “Today’s like Mexico…!”
That was enjoyable. Thank you.
Streets of Bakersfield with Dwight Yoakum and Buck Owens.
Dwight Yoakam & Buck Owens ~ “Streets of Bakersfield” - YouTube