Nope. With Ditto, once a Pokemon is determined to be Ditto, the servers make it Ditto for everyone. With Shiny, it's done on a player-by-player basis.
That means you and a friend can both catch the same Magikarp, yours could turn out to be Shiny, and theirs... not.
With one exception so far: For the July 2018 Community Day, Pokemon Go had Sunglasses Squirtle available through Field Research and, tasks that had a Shiny Sunglasses Squritle were Shiny for everyone!