To me, there's a slight difference.
I check every story. -- I examine every story in detail, typically to look for errors.
I check on every story. -- I examine the situation regarding every story, which may include looking for errors, but also covers checks on how the story is developing/progressing. For example, if I was a newspaper editor, I might say "The first thing I do each morning is check on every story". This doesn't necessarily mean that I get involved in the detail; it could just mean that I go round the office and ask each of the reporters how they're getting on.