Vicious dog accompanying a Republican judge (7)
I believe the answer is:
arbiter
'judge' is the definition.
(I know that arbiter is a type of judge)
'vicious dog accompanying a republican' is the wordplay.
'vicious dog' becomes 'biter' (I've seen this in another clue).
'accompanying' means one lot of letters go next to another.
'republican' becomes 'R' (abbreviation in US politics).
'a'+'r'='ar'
'biter' after 'ar' is 'ARBITER'.
(Other definitions for arbiter that I've seen before include "Rarebit? (anag)" , "One has the authority to resolve disputes" , "Settler of dispute" , "He'll decide" , "Mediator or referee" .)