As someone who was an RA, EMT, and was stalked by a patient, I feel like I could weigh in here.
I don't give my last name on calls unless asked. It's not due to fear or anything, it's just because it's rarely needed. If asked what my last name is I answer--it's on my uniform shirt anyway. Every ambulance company I've volunteered or worked for has listed all the staff on the website anyway, so it would be child's play to Facebook search through the handful of "Nicks" on the roster and find me. Like, for example, what a certain college student did to me after she fell for my roguish good looks on a call. She then proceeded to call 911 fourteen times in an attempt to see me. It was an interesting month to say the least.
But it would have happened even if I told her my name was Artichoke, she could just request a copy of the run report. Or just kept calling 911 until I showed up again and read my shirt.
At the end of the day, it's a risk we take in healthcare, just like violent patients. She ended up getting counselling and it all ended fine.
As an RA, I don't see why that information is needed, and you're certainly not legally required to disclose it to the RA. I always just wrote "Student XXX was transported by YYY EMS to ZZZ hospital" and that was fine for our Student Life dept. That being said, if any legal action is going to be taken your name is on the run report so is it really a big deal?