The most common and most likely answer to this crossword clue is the 10 letter word FREEMASONS
I've seen this crossword clue in The Times
F
R
E
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M
A
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More clues leading to the result FREEMASONS
Buddy explains!

The straight part
We know that "private club" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case private club = freemasons.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 FREEMASONS mean?
FREEMASONS noun- A member of a guild of skilled itinerant masons during the Middle Ages and early modern times.