According to answers on a Monk mantra question , the following should also be true of Shouts:
Different shouts
do stack (you get the effects of all distinct shouts)
Multiples of the same shout do not stack Differently-runed versions of the same shout will not stack, but you will get the effect of both runes.
gaming.stackexchange.com/users/3241/Wikwocket
Tater596, you are partialy correct.
Shouts do not stack, however, if different runes are used, they do.
If for instance 2 barbarians use their shout one runed with impunity (50% resistance bonus), the other with invigorate (+10% maximum life, 310 life per second), both would gain 20% armor, 50% resistances, 10% max life and 310 life per second.
War cry does in fact work the same way monks mantras do.
gaming.stackexchange.com/users/28740/SergeantMad
Effects of the same kind do not stack with each other(unless stated in the tooltip, like with Frenzy).
Casting the same shout with the same rune will merely refresh the timer of the current effect.
gaming.stackexchange.com/users/1155/Mr. November
I just tested all of this out, and here is what happened:
Me and and another barb, both with War Cry, used our shouts together.
He had Invigorate, I had Impunity.
The runes do not stack.
When he used his shout, I got the life.
When I followed up with my shout, I gained resist, but lost all the extra life and life regen.
Same shouts do not stack, regardless of what runes are in them.
gaming.stackexchange.com/users/33380/Kevin
As stated above by Mr. November, these shouts are friendly buffs and will not stack regardless of the runes attached to them.
I'm not sure if you are/were a WoW player, but think about the shouts that could be done by different warriors.
If two warriors were in a raid, they would generally decide which shout would be done by which warrior.
One would generally do "Battle shout" to increase attack power, and the other might pick "Commanding shout" to increase stamina.
Each of these warriors might have talent points invested in improving these shouts, and they would generally let the one with the most improved shout cast it.
It is basically the same in Diablo III.
gaming.stackexchange.com/users/25581/Tater596
Do Barbarian shouts, such as  War Cry , stack? If two barbarians cast War Cry, does the armor bonus stack or does it remain constant at increasing armor 20% for 60 seconds? What if each Barbarian has a different rune equipped?
Do Barbarian shouts stack for party members?
gaming.stackexchange.com
<diablo-3><diablo-3-barbarian>
gaming.stackexchange.com/questions/66648
gaming.stackexchange.com/users/1770/Elpezmuerto
Shouts do not stack if different runes are used. Effects of the same kind don't stack with each other. Shouts do not stack, if it isn't specified in the skill's tooltip.
Players explaining that usually shouts don't stack, except it is detailed in the skill's tooltip.