As of release, there do not seem to be any shiny Pokémon in the game. Although it's hard to prove for sure since the game's code isn't widely accessible yet, I haven't found a single report of any shiny encounters.
Since the way the game works is that each of the Pokémon is supposed to be a unique friend with unique abilities, with no way to repeatedly meet more members of the same species, I would speculate that the encounters are more or less hard-coded, since they're much closer to narrative characters than random spawns.
That doesn't guarantee that there won't ever be any shiny Pokémon in the game—it's possible that they'll add a character whose special trait involves being shiny, in line with the other special Pokémon variants like DJ Rotom. Or there may be some other type of randomized event or mini-game where you can hunt for rare outcomes.
All we can say for now is that the Pokémon you normally meet during the game do not seem to have the random possibility of being shiny.
Source 1 / Source 2 / Source 3
