LANGLEY, Va.—The President’s Daily Brief is among the U.S. government’s most closely held intelligence reports, its highly classified contents intended only for President Trump and a handful of his top national security aides.

Lately, an extra—and uninvited—pair of eyes has been looking in on the PDB, as it is known, even before Mr. Trump gets to peruse it. They belong to “Hudson,” a red-tailed hawk,

Many...