I'm almost positive this applies to every university (HIPPA), but I know at my university, they don't and can't tell anyone about anything you discuss in a session unless you're planning on harming yourself or someone else. I've been seeing a therapist at my university's counseling services for free for 2 school years now. 99.9% of the therapists are LCSWs (there's one clinical nurse specialist), so they don't do med management/chronic treatment period, but I think technically they're supposed to limit the amount of sessions you go in for. There was talk of a wait list (I was put at the "top" because I was physically ill and on the verge of not being able to function from my anxiety and was referred from the school's health services and when I asked my therapist about this "list", he did say there was one). But this mysterious list stopped being mentioned eventually, and I was never told that I had to stop.

I don't know if my therapist was supposed to let me keep coming in, but I've been going on a consistent every other week basis, and it was even at every week for a month or two this past semester. Granted, I go to a relatively small university (~7500 students), so it might not be as big of a deal. As far as my experiences, I got lucky and was placed with a therapist who has been perfect for what I needed. It has been more than helpful for me. Unfortunately, my school decided to stop hiring contract therapists next year, and he was one of them (as are like 5 or 6 others). So there's always that chance of sudden termination with little warning. It shook me for a few days and then I was fine, but I know some people become pretty attached, so that's a strong negative for relying on campus services.

At the very least, you should try. If they do turn you away, ask for suggestions or referrals to therapists in the community based on what you need and can afford. And I really wouldn't worry about being prevented from going on study abroad unless you're suicidal or homicidal. They have to maintain your confidentiality and won't share anything without your permission, and it's hard to think of a reason they would want to share anything.