While I've been a professional for over half a century, (those Sackbuts were a challenge, eh Franklin?), and worked in many countries, I've never known anyone with perfect pitch. Relative? I worked with another arranger in D.C, and when we were bored, we used to get together and transcribe chords. One dictated and the other transcribed. We transcribed not only complex chords but twelve-tone stuff, as well. Both of us had relative, not perfect pitch.
Both of us were also multi-instrumentalists. I've noticed that an additional dimension takes place when going from transcribing from air waves to playing a horn, and that is the acoustical properties of the horn are added. You may hear a pitch perfectly, but different registers, pitches, etc. may alter that pitch and you have to adjust to produce the true pitch, (and also adjust the pitch to fit in with the surrounding chord tones coming from other horns), by using alternate fingering, humored slide positions and the like. Hearing, then, comes into play when adjusting.
You don't need perfect pitch, but good relative pitch combined with a sensitivity to how to adjust your horn to the pitch are needed.
p.s. I knew a of a sax player who had perfect pitch and he could only play C Melody sax. Transpositions drove him crazy and he was very uncomfortable even when playing with other players. I have a friend in the Philadelphia Orchestra. I asked him one time, "How do you guys play so well in tune", and he said, "We don't play in tune. We play out of tune - together", LOL.