Romals are a West Coast/Californio tradition. The closed rein and way of holding the reins does seem to help a horse stay more square and upright in their shoulders. They are, traditionally, used on a true bridle bit as the last stage of progressive bridle-horse training. They are also used in modern cow horse, trail and sometimes reining competitions among stock horses, and are commonly seen on western show horses in the Arabian and Morgan breeds.
You want your romal reins to have some weight and “life” to them. The weight is because they are designed to allow the horse to work off of a signal rather than an actual pull on the bit. The horse can feel you lift the reins before you ever make contact with the bit. Lightweight romals are pretty worthless, as they don’t work as intended.
For regular showing, your romal reins are held in the left hand, thumb up. Quirt is held in your right hand. Different associations have different rules regarding the amount of space/slack that must be between your rein hand and the hand holding the quirt, and whether or not the free hand can be used to help adjust your rein length by pulling or feeding rein through the rein hand (generally, no for pleasure and reining, but okay for working cow horse and trail, always check rules, though).
If you’re doing a right hand trail gate, or roping with romals, the proper way to handle them is to pass the quirt over to the left side of the neck, so that it’s out of the way. You hold both reins and quirt in your left hand (doubled at the connection, so both reins and end the quirt are coming out the bottom of your hand. Your right hand is then free to work the gate, rope, pony a youngster, or whatever. This is also the proper way to hold the reins for mounting and dismounting-- gather reins in left hand, mount, then pass the quirt over to the right side before riding off-- reverse to dismount.
I really like riding with a good set of romals, but I first learned to ride with them as a kid, before going to splits for quite some time. I’ve been quite happy to see the Neo-Californio movement take hold, and I’m happily showing trail horses and riding in romals again.