Yes, news media can safely publish information about student-athletes without violating the law.
However, publishers may run into some legal difficulties based on the sources of their information.
The First Amendment, with very few exceptions, protects the publication of truthful information. Even when somebody engages in an unlawful act, such as illegally recording a private cell phone conversation, the publisher is not going to get in trouble for broadcasting or printing this information — as long as the publisher had no part in the illegal activity. This was the case in the U.S. Supreme Court’s decision on this very situation, Bartnicki v. Vopper, 532 U.S. 514 (2001).
As long as the information published is newsworthy and isn’t an outrageous and highly offensive violation of a person’s privacy — which are more typically personal medical or sexual matters — then the publisher won’t open itself up for civil litigation on privacy grounds, either.
That said, the source of this information could run into some legal problems. The Federal Education Rights and Privacy Act (FERPA, also known as the “Buckley Amendment”) protects the privacy of student educational records. Schools that receive federal funds can lose funding if they violate the provisions of FERPA, and the U.S. Department of Education can investigate unlawful release of student records.
Records of school-related drug tests would certainly be protected by FERPA, and any school official releasing these could be in violation of the law. A publisher receiving this information can legally publish it, of course. But that publisher may very well be called upon to identify the source of that information under federal subpoena. With no federal reporter shield law in place yet, publishers would have little choice but to reveal the identity of the source or face contempt of court sanctions.
The best way to deal with this would be to get sources on record that are not linked to the school administration — the players themselves, teammates, parents, or anyone else with specific knowledge of the reasons for their dismissal from the football team.