Does party size affect chance of better loot?
When a monster is killed and an item is dropped in multiplayer, does everyone get the same item? Or does the item get randomized for everyone.
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It is randomized for everyone. The items are totaly dependant on your session. While you might get a rare from a monster the guy you play with might only get a regular item.
The items that drop from monsters are randomized for everyone (like the D3 tutorial says, "everyone gets their own loot", but depending on your own Magic Find / Gold Find, you may acquire better items ranging from Magic to even Legendaries.
However, if you don't want to farm and wait for a Rare or Legendary (Magic is pretty common) to drop, or one that randoms you a good amount of stats, you could always visit the Auction House. Trading loot is one way to improve yourself as well.