The most common and most likely answer to this crossword clue is the 8 letter word SANCTUMS
I've seen this crossword clue in Independent
S
A
N
C
T
U
M
S
Buddy explains!

The straight part
We know that "private rooms" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case private rooms = sanctums.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 SANCTUMS mean?
SANCTUMS noun- A place set apart, as with a sanctum sanctorum; a sacred or private place; a private retreat or workroom.