I think people have generally understood the question to be about the legal authority of the office but I think there are interesting implications of not communicating or recording a declassification decision.
From a practical administration perspective:
"procedures should have been in place to notify the official who originally classified that information who in turn would have to have notified the potentially millions of individuals who derivatively classified or otherwise had copies of that same classified information.
"[Without such procedures] the original classifier and the myriad of authorized users of that information would remain oblivious to the fact that the information contained therein would no longer have the legal protections of the classification system." - J. William Leonard, former Director of the Information Security Oversight Office (ISOO), 30 years experience in that world
Of course there are hitherto unauthorised users who would like to be aware of the material's status as well, e.g. journalists and academics.
In terms of the president's legal authority while in office, apparently he doesn't have to communicate his decision.
In terms of what may happen with legal proceedings if, after he left office, the authorities were to search his property and find what looks like classified material (drawing from current events):
he claims he declassified it while in office
but there is no evidence he followed the well-established declassification procedures with respect to this material (there is evidence he followed the procedures with respect to other material)
there is no evidence of any change or exception to those procedures
there is no evidence of a relevant declassification decision, memorandum, executive order, policy or "standing order" (that "documents removed from the Oval Office and taken to the residence were deemed to be declassified the moment he removed them")
there is evidence the material recovered from his property is classified - the labels on documents and boxes that says "classified/TS/SCI", any extant 'paperwork', copies of the material held elsewhere, digital records stored by the government to that effect
earlier in the year, after a previous visit from the FBI, his attorney signed a statement that said something to the effect of "we have now returned all the classified material" (implying there was classified material to be returned, and a false statement to the FBI if some remained at the property)
the current government may say "this material is in fact classified" and have evidence (copies, paperwork etc) that it is
It's his word vs. a lot of evidence to the contrary.
So yes, the president may have the legal authority while in office to declassify material without filing paperwork but if there's no evidence of his decision there are contemporaneous practical and legal problems and, if he took the material home, there are post-office legal risks.