ZDoom

Currently there is no way to do that. You can try asking for a feature suggestion, though - this would not be hard to add.

Is there a way to disable staying of weapons and keys in Coop?
We want a weapon to only be picked up by one player, so one has to chose who get which weapon.
Additionally, the levels we play have triggers activated when a key is picked up, and when a key does not disappear it is not picked up, and thus we cant complete levels in coop (which works flawless in Single player).