1) In the first fanfiction, I'm pretty sure the story takes place after the Avengers and the OC was a friend of Jane Foster, she was taken with her on Asgard (but I think it wasn't intentional). Odin basically didn't want her here (she said something that didn't please him I think) so she was treated as a guest but also a prisoner and couldn't leave her room, and she discovered Loki's room was next to hers. He was forbidden to leave his room after the events of the Avengers but wasn't kept in a cell like in The Dark World.
I remember she tried to leave the room by the window but there was a shield to prevent her to do it and Loki was here because they were neighbors and he basically said she was stupid or something to do this.
They met in secret a few times in his bedroom to talk, also I think we discover something about her, and that's why Loki was talking to her. I don't remember the ending but I know it was completed.
2) In the second fanfiction, Loki fell on Earth after the movie Thor and was stripped from his powers, he was forced to work with SHIELD and there was an OC on this fanfiction that didn't like him very much at first, but they had to work together. I remember she was close to Phil Coulson and also that he died in this fic after some attack on their base or something like that. That's all I can remember. I don't know if it is a WIP or if it is completed.
Thank you for your help !