i'm with you--no one else should read one's journal. i live alone so it's not a problem. i also throw out old journals now and then because for me journaling is usually just a way to get out my negative emotions. i wouldn't want anyone to read what i wrote about them when i was upset!
there aren't any rules as to how to journal. if it works better for you to do it on the computer then that way is just fine. also, as was mentioned you can ensure your privacy with a password. just make sure it's something no one will think of. happy journaling!