Letter tiles arranged by each person, initially (7)
I believe the answer is:
epistle
'letter' is the definition.
(epistle is a kind of letter)
'tiles arranged by each person initially' is the wordplay.
'arranged' indicates anagramming the letters (letters rearranged).
'by' is a charade indicator (letters next to each other).
'initially' suggests taking the first letters.
The first letters of 'each person' is 'ep'.
'tiles' anagrammed gives 'istle'.
'istle' put after 'ep' is 'EPISTLE'.
(Other definitions for epistle that I've seen before include "Such as St Paul wrote" , "a for instance" , "Missive; letter" , "A letter from St Paul" , "Paul's work" .)