DoTs do not stack. If you reapply a DoT before the counter is up, you simply refresh the counter. However, this only applies to YOUR DoTs.
An interesting case where this doesn't fully apply (although this is not necessarily a DoT) is the Demon Hunter's "Bola Shot" ability. If you hit an individual enemy with 3 bola shots, you will get 3 explosions.
Following your example of two monks both using exploding palm, each monk would get his own instance of the DoT. I can say this from experience as my friend and I have played through to Act IV in a co-op both as monks. We both tend to use exploding palm on boss fights, and can see the HP counter above the bosses head accelerate dramatically between No DoT, 1 exploding palm, then application of a second exploding palm.
I couldn't find a blue post totally confirming this, but from the observation of my friend and I through four acts, dots of the same type but from different players apply their own effect for each player.