I read this comment earlier (quoted in case it's removed, emphasis mine) and If a Pokémon is shiny for someone, is it shiny for everyone in the same location?:
I also can confirm this: I just had a regular Magikarp on the map. Once the battle started it was yellow and had sparkles. So yeah, you need to invoke the battle. It wasn't shiny for another person, though
The same Pokémon was shiny for one person and not another. This surprises me, because Pokémon are otherwise the same for all: same IVs and same CP for people on the same level (this has been my experience, playing with my partner of the same level; however, this does not appear to be other people's experience)
What determines the shiny nature of a Pokémon? Is it determined for each player when it spawns, or when a player clicks the Pokémon? Does running and re-entering affect this (i.e. does running from a shiny and reclicking it keep its shininess)?
Answering the last question should be enough to answer the first two.