From the beginning of the novel, Jem sees Boo as the boogieman of the neighborhood, as do all the children in Maycomb. He is the main influence on Scout at this point. He has all of the stories, whether they are true or not, and he makes sure to tell them all to Scout in detail. Although Jem slowly makes the realization that Boo is just trying to be friends in chapter 7, Scout never catches on to this.
Scout does know that Boo put the blanket around her shoulders in chapter 8, but she still doesn't see it the same way that Jem sees it. Jem is still trying to figure it all out, but he doesn't come out and explain it to Scout. In fact, it takes two years for Scout to finally see Boo as a good neighbor--one who saves their lives. Maybe some of what Jem has told her over the years sinks in, but it's not evident until chapter 26 when she admits that she felt remorse after "having taken part in what must have been sheer torment to Arthur Radley." She still fantasizes about seeing him on his porch and talking to him like it was any other normal neighbor. And she of course shows her complete understanding of the sacrifice Boo made when he saved them when she stands on his porch in chapter 31.
However, Jem's influence only went so far. The book's focus was on Scout and how she came to realize life's lessons somewhat through Jem, but it was mostly through the words of her father Atticus. She always noticed Jem's moods, and she seemed to understand to a point why he was upset from time to time, but most of her feelings about Boo were because of the lessons she learned from her father.