A purebred Shagya is not pure Arabian blood; in fact, they can’t be more than a percentage of purebred Arabian. The original breeders in Hungary crossed Arabs imported from the desert with native mares to create a warmblood type breed with a strong Arab influence.
In North America, the only way to restart the breed was for the Europeans to approve a select group of Arabian mares that followed the breeding of the original Shagya lines in Europe crossed with an imported Shagya stallion named Bold Bravo (and others), who was found by Adele Furby just days before he was to be put down at the bequest of his deceased owner, a Hungarian. Later, the American breeders used imported semen to add to these lines.
A Shagya is going to either be a very sturdy warmblood type, or they are going to be more of an endurance type (generally lines tracing back to a stallion named Oman, who competed in the Tevis Cup.) The stallion lines with the name Shagya in them look and move more like warmbloods. Adele Furby breeds the warmblood type, although she did have a couple of the endurance types standing at her farm for a while.
Personality wise, they tend to be either very laid back or very reactive. if you get a smart reactive one, they are not the easiest ride in the world. My experience is they tend to challenge the rider, regardless, and have a mind of their own. They can be nice movers but the purebreds never seem to be able to compete at higher levels in the dressage world, but they do excell at endurance, and adding them to sporthorse lines really adds a lot of quality. They are easy keepers with good feet, and are for the most part a very pretty horse.