> Congress shall make no law... abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.
Press implies covering events and publishing about them. If that's not convincing enough, they spell it out in right to assemble, anyways. A press corps can assemble to cover events.
I don't think it implies that covering events has some kind of immunity. It's still subject to any general restriction on the assembly. The government can impose restrictions on the time, place, and manner of peaceful assembly, provided that constitutional safeguards are met. See https://www.loc.gov/law/help/peaceful-assembly/us.php