Shiny Pokémon work mostly the same in Pokémon Go with a few exceptions. First, the shiny rate for these Pokémon is not a universal rate like the main series games. For most species, the Shiny Rate is 1 in 500, but some Pokémon have a increased or decreased shiny rate. I won't go into all the details, as that is in my source(1) if you wish to read it. Secondly, Shiny Pokémon aren't the same for each player like you noted. You can encounter the same exact Pokémon, but the shinieness is determined separately for each player. This explains why you and your mom can encounter the same Eevee and it be shiny for you, but not her.
Now, onto when the shinieness is determined. There a 2 major theories on this. Based on what I have read, it is kind of likely that it is predetermined once the Pokémon's ID is generated. Now, this isn't just it. It apparently works as a function of your User ID and the Pokémon's ID. This theory is plausible, but some people disagree with this. Look at my Source(2) for a better understanding of it. I can't explain this well as it gets into math and programming(I think?) That I don't understand.
Now, some people disagree with this theory like I stated above. They reason that Pokémon Go wants to take as many measures as they can to prevent cheating on their games. They believe that it is just RNG when the encounter happens that determines the Shinieness. Without writing 10 whole pages on this, they believe Pokémon Go uses RNG for a couple reasons. Firstly, it would be possible (not easy, but possible) to Reverse Engineer the User ID and Pokémon ID theory to find a correlation between the 2. In this scenario, they find a correlation between the Pokémon ID and a user ID and it becomes possible to literally "Develop a map which shows you your individual shinies."
Conclusion - While my source is 2 years old, it is still true today. I put more merit into the first theory, but the RNG theory can be just as true. Both of these theories have evidence to support them, but nothing conclusive. source Source(1): https://m.bulbapedia.bulbagarden.net/wiki/Shiny_Pokémon_(GO)
source 2: https://www.reddit.com/r/TheSilphRoad/comments/7a4ffq/when_are_shinies_determined/