To achieves these goals, IJM relies on the help of staff from different backgrounds with different sets of skills. These peope are united by a passion for protecting and helping the poor and weak.

The above is from an article named "International Justice Mission." Is it possible to delete "sets of" without missing out anything? Thanks.
Well, of course you miss out something, but I don't think it is very important here.
Teachers: We supply a list of EFL job vacancies
Thanks, Mister.

Does my version still sound idiomatic to you?
I think we Chinese are not used to the original version "different sets of skills?"
Yes, it is OK, but you are missing out on a very natural English expression.