I honestly don't remember... probably somewhere between 10 and 12... it's just the way we were raised. We did things alone all the time without ever thinking about it. Except for swimming. -- no come to think of it, we even went swimming alone, but we always had to tell that we were going. Like Smrobs, we were raised on a need-to-do basis. If I would have always had to have a buddy, I wouldn't have been able to do very much of anything, whether work or play. Just wasn't the time available for partnering-up. Of course, with that type of upbringing, the responsibility level is usually much higher and taught much sooner.
I think that the decision to allow children/teens to do things alone depends HUGELY on their ability, maturity and trust level (and their own desire). Just looking at my two sons I can see the difference in the tasks I trust each of them to do alone at what age. My now 17yrold I would have let drive my truck at 11 down the road; my current 11yrold? Well, I worry about him driving from my lap down a long driveway. In this case, it's mostly skill level that is the contributing factor.
As for adults, we all have our own comfort zones. I truly cherish the times I have alone in the bush, with or without a horse. But not everyone gets re-juiced that way.