There is nothing in the Constitution that gives more free speech rights to the “press” than anyone else.
The First Amendment codifies rights of the people - the reference to the freedom . . . of the press does not in principle exclude any person, whether or not they own a press yet. In principle, you and I own presses, inasmuch as we both have computers with Internet access which enable us to make our opinions accessible to people worldwide. The development of that technology was promoted by the provision of the Constitution which provides (Article 1 Section 8) thatThe Congress shall have power . . . To promote the progress of science and useful arts, by securing for limited times to authors and inventors the exclusive right to their respective writings and discoveries . . .and in that sense the technologies of telegraph, radio, TV, and computer are anticipated in the Constitution.To the extent that freedom to use such communication technologies needs to be limited, that is provided for in the Constitution, too - see Article V, Amendments.