That is something that tends to be done more on the more versatile and trained horses. Generally many of these people use bits or hacks on theirs, and with more practice and understanding of whats being asked by rein pressure or other cues, then its done a little more often in the halter or on just a circled line.
Regarding halters, this is back to front from a lot of NH philosophy, as well as some conventional ones. The horse is started in a halter, and only moves on to a bit when it is educated enough for it to be used for communication rather than to force control. *True* hackamores(principally the same as a rope halter were actually designed for starting horses.
I believe in the above principles and think it's important to teach a horse well before attempting to use something potentially painful on it. There is virtually no chance of the horse avoiding painful bit pressure(and it causing other negative effects) if they are bitted before they've learned how to yield to pressure in the first place.
Some horses can be controlled more easily in a bit, especially if they haven't been taught to yield properly. But by the same token, using pain, or fear of pain to control can backfire, if it all gets too much for the horse, if he's confused & constantly fighting the pain, refuses or 'evades' the bit, if he wasn't desensitised to it properly & becomes a head shaker.... etc. While a horse that has only been ridden in a bit may need a few lessons before understanding the halter, I've had success 'rehabilitating' 'problem' horses purely by removing the bit, and I'm by far not the only one. It seems the pain or fear of it was their biggest problem & when that was gone, they were suddenly 'well behaved'.
However, riding with just a string around the horse's neck is generally a demonstration of more advanced work. There aren't so many who start horses free or with only a string(look up Nevzorov for one) but they progress to there.
Very basically, you are teaching the horse the same behaviour(yielding to pressure) but the pressure is on the nose rather than the much more sensitive mouth. The riding & cues can be essentially the same. You should be teaching the horse to yield to your body, legs, then rein generally, so that they begin to anticipate & you can do less. Obviously you want to start in an enclosed area, for safety, but also to give the horse a chance - you can *ask* for something, back it up *politely* with your reins & persist until you get the behaviour, rather than having to attempt to use force to control the horse. Once the horse is reliably responding to your body cues & you don't really need reins, then you can start(again in safe area, perhaps with halter as well for emergency) with the string around the neck... or nothing at all.