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.
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.