When people are at the point of applying to veterinary school, they are adults. They may be young, but they are adults and should act like it. That includes conflict management and conflict resolution on their part, not immediately going to "an adult" for help when they havent done anything to fix the situation themselves.
If this were me, I would approach this individual and say, "Hey, I heard you copied my personal statement. We need to tall about this." That gives them the opportunity to fess up and rectify the situation. If they double down and deny it, then I would ask to see it. They dont give it up for me to see, then I would go to the clinician and ask for help, explaining I had tried to fix the situation and had been met with resistance.
As a clinician I would absolutely not want to be the one managing the situation unless absolutely necessary. Going directly to the clinician deprives the students involved the opportunity to be adults and grow from the situation.
For what its worth, based on the format of the "personal statement" being divided up into three separate sections that are just so short, I think it would be easier than people anticipate to steal a personal statement than they think by changing things here and there. Instead of the eureka moment coming from helping a tiger in a zoo, its helping a horse on a racetrack. Instead of their mom being their inspiration, it's their childhood best friend. Instead of battling cancer as a teenager/undergrad, its depression or an eating disorder. Etc. Etc. As a whole, we're a clever lot, and the tone/intent/whatever of writing can be pretty easily manipulated, especially when a completely different person with completely different perspectives looks at it.