The most common and most likely answer to this crossword clue is the 8 letter word INTERNAL
I've seen this crossword clue in The Daily Telegraph
I
N
T
E
R
N
A
L
More clues leading to the result INTERNAL
- Fiendish foxtrot gives way to tango from within
- Blessed with time for fine home
- Knight in messy latrine located within
- … Bury facing new wingers from Arsenal inside
- Private and fancy aren’t in line
- Dashed, needing time for female domestic
- Domestic bird seized by half-cut criminal
- Domestic name for velocity in space
- In time, learn to unwind inside
- Trainee best left inside
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 = internal.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 INTERNAL mean?
INTERNAL adjective- Inside of something
- Within the body
- Concerned with the domestic affairs of a nation, state or other political community.
- Concerned with the non-public affairs of a company or other organisation