Well, the game already has this function (AI shouting: "Soldier is history" et cetera). But here's another (simple) way to do it:
In the init line of the enemy units, write: this addEventHandler ["killed",{confirmedKill=true"}]
Then make a trigger and set it to repeatedly
CONDITION: confirmedKill
ON ACTIVATION: leader player sideChat "I got him!"; confirmedKill=false
Or if you want to make it a bit more uncertain, change the init line to this addEventHandler ["hit",{confirmedKill=true"}]
And make the trigger look like this
CONDITION: confirmedKill
ON ACTIVATION: leader player sideChat "I got him! I think he's dead."; confirmedKill=false
That's a very easy way to do it. It's fairly easy to add corresponding soundfiles to this trigger based communicationssystem. The downside is that it will be the leader of your group who calls out every hit.
But if you study the evenHandler section of the comref carefully you will see that the eventHandler function also returns information about who hit/killed every unit. Thus it's possible to make a script that makes the soldier delivering the killing blow call out every hit. But it takes a little more scripting to do, and it really doesn't add that much to game play. Good luck!
Syntax is not completely guaranteed, btw. But I think the eventHandlers look like that. Check the comref to be sure.