The most common and most likely answer to this crossword clue is the 10 letter word ONE-TO-ONE
I've seen this crossword clue in The Guardian
O
N
E
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T
O
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O
N
E
More clues leading to the result ONE-TO-ONE
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 = one-to-one.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 ONE-TO-ONE mean?
ONE-TO-ONE noun- A personal relationship between two people.
- Matching each member of one set with exactly one member of another set.
- Involving direct communication between two people. Used especially to refer to situations in which one person is teaching one other person.
- (of a function) Injective, being an injection: having the property that no two elements of the domain are mapped to the same image.