When I was active on the internet, almost everyone I knew saved their old IMs/PMs (instant message/private message, depending on which service you used). It was easy to do in your 'preferences.'
And when I was active, nobody I knew saved their IMs. When I was in a position to have to save an AOL log (due to an office situation) nobody knew how to do it. And I'm guessing kids especially would just as soon not leave that kind of diary for their parents to review.