From what I understand of the loot system, there are four drops, and all 8 players can roll on any item. This means that for a single run you have the following odds of obtaining at least one item, provided that you, and every other person roll on every item.

P(Get at least one item) = 1 - P(Get No Items)
P(Get at least one item) = 1 - (7/8)^4
P(Get at least one item) = 1 - .5861
P(Get at least one item) = .4139

So by changing the system you have only increased your odds of obtaining any item in a run by approximately 9%.

Since, the math shown by both of us only applies to receiving any item and I don't really know the specifics of the system, I can't comment on how much it would speed up you obtaining that one drop you need for whatever piece it is you want.