I haven't tried multi-player yet, but I know people can drop in or out. If they left items in shared stash, I wonder if other players could use it. I can image such a system being abused.
I'm fairly sure everything in shared stash has always been a free for all in regards to multiplayer, or in theory it is.
My proposal is just a separate supply stash that takes food items, eats them (as in, removes them from the game since it'd be one less thing for the game to keep track of), and automatically tallies up how many units of supply you sent in. The only way to deduct this number would be through taking long rests, since presumably anything you send in can't be retrieved back.