The most common and most likely answer to this crossword clue is the 6 plus 4 letter word PARISH PUMP
I've seen this crossword clue in The Times
P
A
R
I
S
H
P
U
M
P
More clues leading to the result PARISH PUMP
Buddy explains!

The straight part
We know that "Of local interest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case of local interest = parish pump.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.
What does PARISH PUMP mean?
PARISH PUMP adjective- Parochial, of local interest only; petty