2a. Peer editing academic papers require critical-thinking skills and diplomacy.
2b. Peer editing academic papers requires critical-thinking skills and diplomacy.
Your subject is "peer" editing academic papers modifies peer. Peer is singular so which one would be used with a singular subject.
I have a different interpretation, but the result remains the same. "Peer editing" is the subject, with "peer" modifiying or particularizing "editing", even though "peer" is a normally a noun. In any event, the subject is singular, so 2b. is the answer.
"Academic papers" could (or should) be preceded by the word "of", and is essentially a prepositional phrase, (without a preposition) modifying "review"; so "papers" is not the subject of the sentence.
It's a good question!