Private lives: topless sex club’s regulars drunk (9)
I believe the answer is:
exclusive
'private' is the definition.
(similar in meaning)
'lives topless sex club's regulars drunk' is the wordplay.
'topless' means to remove the first letter.
'regulars' indicates alternate letters.
'drunk' indicates anagramming the letters.
The alternating letters of 'club' are 'cu'.
'sex' with its first letter taken off is 'ex'.
'lives'+'ex'+'cu'='livesexcu'
'livesexcu' is an anagram of 'EXCLUSIVE'.
(Other definitions for exclusive that I've seen before include "Select, high-class" , "Scoop" , "Catering for only the wealthy, not obtainable elsewhere" , "article kept away from other rags" , "catering for a few customers" .)