Intro: The most important thing about this post to realize I guess would be to actually think about it, and read it over before you jump to conclusions and say it doesn't work and can't be re-done.
Basically, I picked this up when I was able to turn in a quest (The vials of eternity, the old attunement quest for hyjal which you must do to attain the Scale of the Sands rep reward rings.) and still have it in my log afterward to go back and redo, and turn in.
The only reasonable reason for doing this be to get a second 3k rep (not likely though as the rep is easy to get, and the xpac right around the corner). However, lets say in the expansion there is a quest that gives a very powerful reward that you could use two of? Try this out and you may be able to get two.
My exp: I was in TK with a few members of my guild not long after I hit 70 on my RaF account. Before I went I made sure to pick up the vial quest so I could get the first half done. After the kill I was not able to loot the body but it didnt feel like I was going to D/C. I'm sure this has happened to others, happened to me a few times. So I logged and upon logging back in my friend thought that I didnt have the vial quest, and shared it with me, but I told him I already had it yet the share still poped up so I decided to accept it just in case and for a second I thought I actually didn't have the quest cause you can't have two of the same quests. Heres the thing: When you get the lag where you can't loot corpses, you can share quests the recipient already has (maybe has already done?)
Just after the kill with two of the quests in my log: http://i34.tinypic.com/veu0lt.jpg
The a few weeks later when an ssc rolled around: http://i37.tinypic.com/2mg121c.jpg
Completed twice.
The turn in: The items will get consumed in the turn in, this will only enable you two go back and do it again.
At the turn in: http://i36.tinypic.com/2q0n8dl.jpg
After turn in and after I got my ring with it still in my log: http://i33.tinypic.com/308wh3k.jpg
The recreation: The following is what worked for me not too long ago with a trash quest.
1. You and your friend get the same quest.
2. Your friend spams the share quest button.
3. You create lag.
4. You get quest.
5. You profit.
Try any method to create lag, unplug your internet (server shutdown maybe like the old vendor trick?), open several ventrilo clients and connect at once (worked for me).
Good luck
oh yeah, no stupid comments about my post in contrib section, stay on topic.