The most common and most likely answer to this crossword clue is the 5 plus 5 letter word BRASS BANDS
I've seen this crossword clue in Financial Times
B
R
A
S
S
B
A
N
D
S
More clues leading to the result BRASS BANDS
Buddy explains!
The straight part
We know that "blowers of their own trumpets" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case blowers of their own trumpets = brass bands.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.