Why can't there be a simple option to have the loot auto assigned and sent to a character no matter who picks it up? An even better solution would be to have a common loot stash where loot went when picked up. Players could be given 10 points for every item that was picked up and they could bid on items in the common stash. The items would have a hidden player assignment so that if the game crashed or otherwise suddenly stopped the unclaimed items would go to those players.
The simple fact of life is that there are people that you could have a blast playing local coop with but are jerks or selfish. It is one thing to play competitively against them within the confines of a game. It is another when a game encourages that competitiveness out in the real world.