Sticky question! I think it should be worded as "if the staff members don't stop picking at each other": that would solve the problem immediately. Otherwise, you have to decide if 'staff' is a collective noun (singular) or a bunch of individuals (plural).
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Yoong LiatAccording to some grammarians 'each other' is used in relation to two people, whereas 'one another' is used when referring to more than two people. However, many writers use 'each other' when talking about more than two people.Hi,