The most common and most likely answer to this crossword clue is the 9 letter word INNERMOST
I've seen this crossword clue in The Daily Telegraph
I
N
N
E
R
M
O
S
T
More clues leading to the result INNERMOST
- Private hotel on lively Rome street
- Cool monster bananas at the core
- Parking out of town? Nearly all is private
- Sort of secrets revealed in Monsters, Inc., each detailed crudely
- Private hotel, second in Queen Street
- In it, modified sermon replaces one that’s extremely intimate
- Corrected mentors in private
- Core energy given by minestrone soup
- Central bar alongside square in bustling metro
- Secret demonstration could be given by this toad
Buddy explains!

The straight part
We know that "Private" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case private = innermost.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 INNERMOST mean?
INNERMOST noun- That which is innermost; the core.
- Farthest inside or towards the center or middle.