Looking to the paper in question the first thing to point out is that it's not a clinical trial or even a cohort analysis, but rather is a case study of just two patients.
Both of the two teens in question chose traumatic diets as a means to fuel their losses.
The first patient, Daniel, is described as having approached his weight loss by means of eating no more than 600 kcal per day while running high school cross country. He also apparently eliminated sweets, fats, and carbohydrates, and ate only "diet food".
The second patient, Kristin, is described as having commit to a dietary regimen of 1500 kcal while running 7 miles per day for 3 years.
The authors of the case studies very sagely point out that in children, weight loss, especially rapid weight loss, should prompt primary care providers to explore the possibility of an eating disorder as eating disorders can present at any weight. In the case of these two teens, their eating disorders first manifested in the traumatic diets they both undertook in order to lose weight. Had their family physicians or pediatricians explored their losses when they began, the severity and disordered nature of the efforts might have been uncovered long before these two teens developed their traumatic-diet-induced psychological and physiological signs and symptoms.
What this paper did not however conclude is that weight loss in teens leads to the development of eating disorders and yet I've seen this references to this paper crop up regularly on Twitter since its publication and wielded by various trusted allied health professionals as proof that weight loss in children and teens is in and of itself risky.
What's risky isn't loss, it's traumatic diets, and frankly they're risky for anyone at any age.
The take home message from these case studies is that primary care providers would be well advised to respond to rapid and/or extreme losses of any patient, of any age, as a red flag suggesting their possible adoption of a traumatic diet, but the simple suggestion of the headlines and the no doubt well-intentioned tweeters, that, "teens who beat obesity at risk for eating disorders" leaves out the all-important qualifier of their traumatic means of losing.