This can be done in a round pen, fenced arena or outdoors if you have enough room and appropriate footing to recover should things go haywire. To gallop a horse completely free of being guided by the reins is a good way to improve your riding as well as the horse's certainty that you can ride them wherever they go, however fast they get there. The way they feel through their whole body can improve from doing this. There's a catch though, and that is that you REALLY HAVE TO DROP THE REINS. I know it sounds crazy, buuut If you run a horse and they're surging ahead of your hands because you're pulling on the reins, you're doing exactly what the jockeys do: helping to keep them engaged. And chances are, they'll just be faster tomorrow.
In a round pen you can even do it with nothing on their head. I put the hackamore on just in case an opportunity arose to use it, which it did right near the end. This little exercise where I'm holding her head around, I'm' waiting for her to step her inside hind foot in front of her outside hind foot (as in a one-rein stop). By releasing for the movement of her foot, I'm hooking the rein TO her foot. I'm creating a conditioned response, a neurological pathway, between the rein and the foot so that later when I'm riding around, maybe she'll steer a little better! The more I learn, the more I'm realizing how much hindquarter control matters to pretty much everything we're trying to teach these things.
Couple of cautionary notes:
1. The reata (rope) should be handled with the same accordance of respect as one would a firearm, as both can get you killed. Until you can handle that thing in your sleep (literally, with your eyes closed) you might exercise extreme reluctance to copy what I did here with the rope. Start on the ground, roping things that don't move, and with a breakaway honda!
2. It's good to do a bit of riding the horse without guiding it, but one thing you want to be aware of in the round pen is that as she looks around to things outside she's traveling around counter-bent as a result. Which they need to learn to do anyway, but go easy on it. Do a little of this and a lot of walking in circles and straight lines making sure they're correct or dropping the shoulder is going to become a habit!
3. When you run a horse, if they haven't been exposed to it much and you're not guiding them either then their self-preservation can begin to get very close to the surface. You've got to be aware of that and manage it, read your horse and know when things are edging close to the line. That line is going to be different for every person but it's a good idea to know yours and work in a way that challenges you without being overwhelming. If you can learn to ride as fast as that horse can run it will make you better but, be aware of where and when you choose to do it, do it sparingly, and again DON'T GET KILLED!