That said, I didn't do anything fancy to stay aware of my breathing minus a few things that you learn in your first couple days (breathe in, breathe out, breath in, breath out 20%). Overtime, you just get a sense of where the rifle tends to point and you learn to control it. Nothing fancy :). It wasn't really a conscious thought process by any means and my coach didn't really push conscious breathing.
The biggest thing he would push was mindfulness of where your shot ended up. He would take away my damn sighting scope and get me to predict where each shot landed.. which was actually incredibly helpful as much as I hated to admit.
The things that were much more important than breathing control were keeping calm (mental state, not necessarily breathing), adjusting your "natural point of aim" after each shot, and having a constant awareness of stance and how small changes affected your aim. Then again, my shooting style had a complete lack of the usual advised technique. My trigger control was garbage, for example. I met up with him after high school and shot for a bit. When he saw my shooting stance, he asked, rudely, "Where'd you learn that faggoty-ass stance from?". "You", was accurate. :)
Funny thing. Apparently shooters and swimmers in their teens suffer from the problem of high pitched noise while sleeping. It's really stupidly embarrassing! https://en.wikipedia.org/wiki/Catathrenia
When I was shooting competitively, I could see the crosshairs move off-target with each heart beat. Squeezing the trigger during a beat would inevitably result in failure.
I've never had to walk off the line because of it (too stubborn), but everyone else on the team did, every other competition.
I used to be very active in competitive freestyle archery (scope and release) for both indoor and 3D. You do work to control your breathing and heart rate but I've never heard of anyone trying to time the release to their heartbeat. Instead many competitive shooters train to disconnect the release action from conscious thought. You literally stand a few feet in front of a target for hours, shooting without aiming, and commit to muscle memory the action of triggering your release. "Back tension" releases are very popular in archery specifically because there is no traditional trigger and they are instead activated by tilting or squeezing.
Once you no longer think about releasing you now focus your shooting solely on aiming. When you're ready to shoot you fire off the muscle-memory-trigger thread in your brain and your body just involuntarily goes through the motions of firing the arrow. When done correctly the actual triggering of the shot is a surprise.
Olympic archery is very similar. Olympic shooters use "clickers"[1] on their bow that tells you when to release. You pull the arrow back and began to aim, continue to pull back slowly until the clicker clicks and that triggers and automatic response to relax your hand and loose the arrow. You practice shooting over and over again such that whenever you hear/feel the clicker you release automatically and therefore separate the release process from active thought.
http://www.k1-archery.com/images/product_images/original_ima...