The most common and most likely answer to this crossword clue is the 4 plus 6 letter word HOME CINEMA
I've seen this crossword clue in Financial Times
H
O
M
E
C
I
N
E
M
A
Buddy explains!
The straight part
We know that "private entertainment" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case private entertainment = home cinema.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.