Would the independent schools know about the other offers children had received? Do the schools talk and if they do does that include information about scholarships etc?

yes they might know as they get in touch with the actual primary school and get a confidential statement from the teacher and head teacher. However, the school can mention grammar school's results if they got them ( Bexley and Kent grammar school do send their results) .
In interview day of my DS, the independent school's headmaster told us that he is aware DS has many options and passed grammar school's tests. none of us hd mentioned anything before; so we assumed that my DS actual school might disclosed this information. I don't think that would affect the child application and they all aware parents would try anything to get the best for their children.