The most common and most likely answer to this crossword clue is the 5 letter word INNER
I've seen this crossword clue in The Times
I
N
N
E
R
More clues leading to the result INNER
- Daughter misses meal — it’s part of the target
- Victor's wife leaving ring circling bull
- Private meal, no starter
- Secret miscreant kept from head
- Secret boozer with beer, half-cut
- Victor losing head in ring
- Victor leaking wife's secret
- Private meal with no starter
- Film director's beginning to go red
- Number one, but not the number one in private
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 = inner.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 INNER mean?
INNER noun- An inner part.
- A duvet, excluding the cover.
- A forward who plays in or near the center of the field.
- A thin glove worn inside batting gloves or wicket-keeping gloves.
- One who supports remaining in the European Union.
- The 2nd circle on a target, between the bull (or bull's eye) and magpie.
- Being or occurring (farther) inside, situated farther in, located (situated) or happening on the inside of something, situated within or farther within contained within something.
- Close to the centre, located near or closer to center.
- Inside or closer to the inside of the body.
- Of mind or spirit, relating to the mind or spirit, to spiritual or mental processes, mental, spiritual, relating to somebody's private feelings or happening in somebody's mind, existing as an often repressed part of one's psychological makeup.
- Not obvious, private, not expressed, not apparent, hidden, less apparent, deeper, obscure; innermost or essential; needing to be examined closely or thought about in order to be seen or understood.
- Privileged, more or most privileged, more or most influential, intimate, exclusive, more important, more intimate, private, secret, confined to an exclusive group, exclusive to a center; especially a center of influence being near a center especially of influence.