Shame detective's good with people (8)
I believe the answer is:
disgrace
'shame' is the definition.
('disgrace' can be a synonym of 'shame')
'detective's good with people' is the wordplay.
'detective's' becomes 'dis' (Detective Inspector).
'good' becomes 'g' (abbreviation).
'with' is a charade indicator (letters next to each other).
'people' becomes 'race'.
'dis'+'g'+'race'='DISGRACE'
(Other definitions for disgrace that I've seen before include "scandalous" , "Bring shame upon" , "Loss of good reputation" , "Shame or loss of respect" , "Loss of honour" .)