The trigger action "Disable multiplayer frag/death messages" also disables death trigger execution for all other players. So if you have a death trigger that shows text in chat and player 2 kills player 1, only player 1 will see the message.
Looking at the code (on line 17681) it seems this also affects other parts of the game, such as exp, disconnecting from teamkilling, enemy death sounds and possibly something else.