Private agent set up new salon (8)
I believe the answer is:
personal
'private' is the definition.
('personal' is associated in meaning with 'private')
'agent set up new salon' is the wordplay.
'agent' becomes 'rep' (short for representative).
'set up' shows that the letters should be reversed in order.
'new' indicates an anagram.
'rep' written backwards gives 'per'.
'salon' anagrammed gives 'sonal'.
'per'+'sonal'='PERSONAL'
(Other definitions for personal that I've seen before include "Individual; advert (column)" , "Of private concern" , "Belonging to an individual" , "Of one's private life - nobody else's business" , "Such a rude remark" .)