Some eavesdropper's on a line getting offensive (8)
I believe the answer is:
personal
'offensive' is the definition.
(eg a personal question is offensive)
'some eavesdropper's on a line' is the wordplay.
'some' indicates a hidden word (some of the parts of the word are used).
'PERSONAL' is hidden in the letters of 'eavesdroppers on a line'.
'getting' is the link.
(Other definitions for personal that I've seen before include "Private - bodily" , "Of private concern" , "Private - (of remarks) offensive" , "Pertaining to the self" , "Individual; advert (column)" .)