The most common and most likely answer to this crossword clue is the 14 letter word HEART-TO-HEART
I've seen this crossword clue in Financial Times
H
E
A
R
T
-
T
O
-
H
E
A
R
T
More clues leading to the result HEART-TO-HEART
Buddy explains!
The straight part
We know that "Conversation of two people" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case conversation of two people = heart-to-heart.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.