TBH it depends on family dynamics and logistics.
In my extended family, my parents always "host" in that events are always at their house as it's the biggest and most equipped for hosting, but everyone contributes.
My parents provide some basic spirits and mixers but everyone brings their own bottles of their preferred drinks and if there's stuff left at the end of the night then they take it home as my parents don't need 5 different brands/flavours of gin/vodka/rum knocking around when they only drink wine and beer. Sometimes if my parents house vodka has been drunk and a guests bottle is almost full then they'll say to just keep it for the next party and it will go in the cupboard but if there's only a tiny bit left in a big bottle then my parents would rather they take it home so they don't have it taking up space when they don't drink it because otherwise they'd probably just pour it out and recycle the bottle after the party.
At the end of any meal/buffet etc, once everyone is clearly finished eating and as people start leaving then they will just automatically pack up anything they want to take with them, though everyone is careful to leave something for other people unless they're told "No thanks, you take it we won't eat it"
We don't have discussions about it because it's just how we do things. My mum might send people home with specific things if someone couldn't attend (Eg on Boxing Day we had 20 people round but one of my teen cousin's was working in their retail job so my mum sent my aunt and uncle home with an extra portion of the roast dinner and a little bit of all the desserts that she had specifically set aside for him) or ask people if they want to take specific things home because she knows that her and my dad won't eat them and the other person loves them but otherwise if it's on the buffet or in the fridge and clearly not "everyday food" then it's fair game to be taken by anyone. The only unwritten rule that you don't take someone else's serving dishes, so if there's only one portion left you check with the dish owner whether they want it, and if not then you decant it into Tupperware.
My parents host because they have the biggest house, they don't want to be left with loads of random food after the fact (especially as they're usually catering for at least 15, but usually closer to 30 for family events) nor do they want to be left with everyone elses dishes to wash/return so it's always well known that nobody leaves a party at my parents house empty handed and you always leave with the serving dishes you arrived with.