Sounds like a private bitterness (7)
I believe the answer is:
rancour
'bitterness' is the definition.
(I know this)
'sounds like a private' is the wordplay.
'sounds like' shows a homophone (sound like).
'a private' becomes 'ranker' (ranker can informally mean a military private).
'ranker' sounds like 'RANCOUR'.
(Other definitions for rancour that I've seen before include "Malicious hostility" , "Hard feelings" , "Bad feeling and resentment" , "Deep-seated enmity" , "Animosity" .)