The most common and most likely answer to this crossword clue is the 9 plus 5 letter word HOUSEHOLD NAMES
I've seen this crossword clue in The Daily Telegraph
H
O
U
S
E
H
O
L
D
N
A
M
E
S
Buddy explains!
The straight part
We know that "public figures" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case public figures = household names.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.