I used to have another 'share' button on the site which only showed Twitter/Facebook share buttons. Only when people would click those buttons would you see how many times the article had been shared. It used a trick by adding an identifier to the link which would update a counter on the plugin maintainer's site to indicate how many times an article had been shared. It would not be able to keep track of how many times people simply cut and paste a link from the article to their share site though. I decided to abandon it altogether as it looked kind of pathetic to have a button on every page saying '1 share' (and everybody knew who did it :)
OTOH, I also observed an article I saw being tweeted from mashable the other day. Unreal, 7 minutes after the article had been published there were like a 100 tweets already. That's just automated tweets and there's nothing 'social' about that. I personally don't fancy such guerilla tactics to make your articles look more interesting, do you?
However it would still be nice if people were to share articles more often as it is, to me at least, a partial indication of how informative articles are, 'how am I doing?'.