Kiwi, I never ever shoot from my horses backs. I ride my horses to where I want to hunt and tie them up. If I see game while I'm riding, I jump off the horse and tie them up or hand the reins to another person. Remove the rifle and step away from the horses before I fire.
My horses are used to the noise. But there are several reason, I'd never try and shoot from the saddle. 1st. Most of my shots are long range. ( see pictures of how big and open the country is) I need to get my rifle as steady as possible to hit a target 300 yards away. No way possible to do that from the saddle. 2nd. I want to protect my horses ears. Muzzle blast from a high power rifle is deafening. 3rd, I don't want to get stepped on. I am focused on the target, I am not watching the horses when I am trying to shoot. I don't want a horse to spook and dance around and step on me. So I make sure I move 10-15 yards away from the horses prior to discharging the rifle.
Unlike cowboy westerns, Most hunting rifles are not the slick little saddle guns seen in western movies. Modern hipower rifles have scopes and are just plain bulkier than the rifles in motion pictures. Getting a rifle in and out of the scabbard is MUCH easier done from the ground, not the saddle.
Taking a few extra seconds to do the above, gives your mind time to digest what it has seen. What was the aminal you spotted, Is it a trophy, Is it an elk or some ranchers cow?
Hunting from a horse is a great experience. But shooting a rifle from a horse's back just isn't part of the adventure. Now I do often carry a pistol while riding in the wilderness areas. It's there in case I run into a bear or lion and want to scare them off. Firing a pistol from atop a horse is possible, But I wouldn't expect to be accurate with it. It would just be more of a loud noise maker, to scare the bear away.