Most Roman Catholics I have known would not do that(throw a Bible in the trash) they would honor and respect it. Back in 1946 or 47 the Gideons came to our classrooms. They carefully explained they were giving a small new testament to each student, but for some reason you did not want one, or if you already had one just keep your hand down. We had a choice, and that was standard practice in the public schools in those days. We had no drugs, guns, nor shooting in those days.
I would not want my children to get their hands on a non Catholic book personally. I have not directly received a bible, but have gotten some pamphlets and defintely throw them away. I don't think that is such a bad thing. I am nice about it and don't do it in front of them at all.