Staff on both sides of a street (4)
I believe the answer is:
road
'street' is the definition.
(I've seen this before)
'staff on both sides of a' is the wordplay.
'staff' becomes 'rod' (I've seen this before).
'on both sides of' is an insertion indicator.
'rod' going around 'a' is 'ROAD'.
(Other definitions for road that I've seen before include "Highway or byway" , "See 10" , "Mine passage; way" , "Way from A to B" , "Path" .)