Its called the African carry.
Slings in the African bush are a big no-no - they just get caught and when you get in the thick of it its the last thing you want.
Carrying the rifle on your shoulder is in fact very comfortable and you can control the direction of the barrels easily - they should down to the ground and to the side.
A good double rifle should have intercepting sears and most do, along with a safety that is often bolted as well - it has a second lever that prevents the safety sliding forwards. Such a rifle will not fire unless safety is off and trigger pulled.
But most of the time only the PH in front will be carrying a loaded rifle. A double can be loaded very quickly and silently when the tracks get very warm. Then the rifle is in hand.
With a bolt action rifle, most clients rifled will be carried with rounds in the magazine and nothing in the chamber.
Saying that, many true African hands, who have also been through bush wars etc always carry a fully loaded rifle and treat every rifle as loaded. They don't rely on mechanical safeties, instead safety is in the hands of the hunter and where the barrel is pointing. If you are in thick bush, pushing a wounded animal the danger from being squashed or eaten is a lot lot higher than the supposed danger of a ND, hence rifle is loaded and ready to go. Buffalo, lion and elephant can move very fast if they want to and you will fractions of a second to respond to a determined charge. Unless the animal is wounded though, most charges are mock charges designed to intimidate - a major part of becoming an African PH or guide is learning to read the animals and not get into this sort of situation. In my early 20's I spent quite a few months in the bush surrounded by big game and we weren't allowed rifles - there were some interesting moments, but at no time did you feel that you needed a firearm of any kind.