Depends on what the custom is with the crowd invited. If you're a group of "everyone pitch in" on every party, no problem. If you specifically invite it as a "pot luck," no problem (but pot luck means pot luck, no assignments). If it's the classroom end-of-year party or the scout troop meeting, fine.
But if I have a party under any other circumstances, I assume I will provide everything. If someone offers to do something, I either say, "Don't bother, thanks" or I ask them what they want to do. I do not assign. Not at all. If the second person offers to bring something, I'll say, "someone is bringing paper goods, so beyond that, what would be convenient for you?" But I would never say BYOB.
If you bring food, I think it stays. If you bring supplies (hot plates, serving dishes, cutlery), I think they go home with you. If you bring a game, it goes home with you - it costs you nothing and I think it's fine for them to offer a vague "if you have something you enjoy, we'd love to share it with you." I go to a group "dessert and game night" every few months - people bring what they want but this is a big group in an organization, not a private party.
If you bring 3 bottles of wine and the group only drinks 2, you leave the 3rd as a gift to the hosts. If you really want to ration things, you leave extras in your car and only go out for them if there's a shortage ("Oh, my. Looks like we're running low. When I stopped at the liquor store, I bought 2 extra bottles for my house, but let me get them now.")
I have friends who give 2 huge parties a year (they have big families, and include a number of neighbors like us). They provide everything and say something like "Desserts and beverages appreciated" but leave it at that - if you bring something, fine, and if you don't, no one (including the host) notices. I usually call her to see if my stash of folding tables, serving pieces, chafing dishes and so on would be helpful. She either accepts or declines, depending on what others have offered. But these are large parties of 50+ people, not small parties of 10 like you are describing.