The most common and most likely answer to this crossword clue is the 13 plus 5 letter word PARLIAMENTARY AGENT
I've seen this crossword clue in The Guardian
P
A
R
L
I
A
M
E
N
T
A
R
Y
A
G
E
N
T
Buddy explains!
The straight part
We know that "one working on private bills" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one working on private bills = parliamentary agent.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.