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Thanks for the question on the History Hub. The White House Press Office is responsible for all release of transcripts to the public and the press. They can decide to release as much, or as little, information they want. Beginning in 1957, the Office of the Federal Register and the National Archives and Records Administration (NARA) began compiling the Public Papers of the President. Public statements, official announcements, and transcripts released by the White House were, and still are, compiled into a multi-volume work called the Public Papers of the President. Here is a link to more information about the Public Papers of the Presidents: https://www.gpo.gov/help/index.html#about_public_papers_of_the_presidents_of_the_united_states.htm. NARA also maintains archived (i.e. not live) copies of White House websites and their press releases. For example here are the links to George W. Bush and Barack Obama.
At the end of a presidential administration, when the records become the property of the people of the United States NARA will hold those records in perpetuity. These include all official and unofficial transcripts of meetings, press briefings, and public events. Assuming of course that those records were created. These records can then be requested by the public through the Freedom of Information Act and under the Presidential Records Act.