Not being able to pick up weapons like the shotgun in the church when you both start a new game and they pick it up and you don’t get it
For mission related objects it happens that you have to pick them up in your game that you are host in and in a game where you are guest let the host pick it up. If the host pick it up and it’s still there then all is good. If not leave the game to pick it up in your world and return to mp game.
If you as a guest pick it up before the host and it’s no longer there for the host to take that’s bad. Then I guess the host is without that item for the rest of the game.