For example if one of Clinton's friends forwarded her a NYTimes story about CIA activity in Syria, and she forwarded that to a deputy, well, now she has both received and sent classified information from her email.
And even though it might have been on the front page of a newspaper, a retrospective review would designate it as classified information.
Would a staff-level person get fired in that situation? I don't know. So much stuff is classified these days, and yet the press is pretty good about ferreting out stories. Any regular reader of the major news operations is probably going to see classified info on a regular basis.
How easy is it to keep track of which public, well-known news stories should be excluded from nonsecure email? I would guess, not that easy.