Can a university provide information about a student's activities?
In the US context, under the Federal Education Record Privacy Act (FERPA), students (whether they're international students or US residents/citizens) have a right to privacy.  This right begins when a student enrolls and continues even after a student graduates or otherwise leaves an institution.  In higher education, the student themselves has the right to privacy.  In K-12 education, the parents of a student have control over the privacy.  

Under FERPA, information about a student consists of "Directory Information" as well as a broad variety of educational records that have stricter privacy controls.  

The exact details of what constitutes directory information vary from campus to campus, but it can include name, address, phone number, email address, dates of attendance, degrees granted, etc.  Students have the right to restrict access to this information, but they must request that that this information is kept confidential.  In response to any request for directory information about a student who has requested confidentiality, an institution will simply not respond.  This can cause problems with potential employers confirming that you graduated with a particular degree and other situations where you might actually want to be acknowledged as a student.  

Your other academic records are kept under much tighter control—under FERPA they won't be released without your explicit written permission. 

So, depending on whether or not you have requested that your directory information be kept confidential, your institution might respond to a request by providing your directory information, or it might refuse to answer the request.  If the directory information is confidential, anyone who inquires about you should get an answer like "I can neither confirm nor deny that John Q. Doe is or was a student at Big State U."   

There are certain exceptions to FERPA privacy rights, including situations where the student could be claimed as a dependent on the parent's income tax return, situations where university officials have a legitimate need to access the information, etc.