D21 is our more sensitive one, D16 not as much. We actually use NPR as a primary news source, and the only time I can remember ever turning it off so my kids wouldn't hear it was on 9/11. Also kept the TV off while they were around that day. And when the Oklahoma City bombing happened, I let them see about 5 minutes of the TV news, then shut it off.
I grew up in a household where kids (esp. girls) were not expected to know about current events. I was woefully ignorant on the topic when I went to college, and had a lot of catching up to do. It made me a bit of a news junkie once I figured out how dumb I was about the world.
So... we have made it a priority for our kids to know about current events. It means answering a lot of questions, and sometimes getting into discussions that are uncomfortable. It means comforting and explaining (as best we can) when something bad happens. But I think overall it has been good for them. For example, when the tsunami hit Indonesia several years ago, my kids came to me and said they wanted to contribute some of their savings to help (actually, they wanted to contribute ALL of their savings, but we settled on a portion of it and I matched it instead). D21 is working at the State Department this semester, and hopes to go into a career in politics or working for State -- if we had shielded her from current events, I don't think she would have found that passion.