I've noticed that some plugins have the version charts displayed on the Stats page while some others do not. What's necessary for the chart to show in the stats page? Also, how does WordPress.org know about the versions currently running on servers around the world?
I believe the the requirements are to have more then one version and a minimum number (not sure what it is) of downloads per each version.
The answer is they don't know, but they do know the number of downloads each version has gotten.
Wordpress knows about the versions running on websites where where wp,plugin and theme version checks have not been disabled programmatically (there is no setting to disable,this has to be done by php code in plugin or theme if one wants to do it).
The existing version of wp, plugins and themes is passed to wordpress.org to check for any updated versions of each of those.
If you have a cron manager plugin (I use mine amr-cron-manager), you will see the cron upgrade checks run twice daily, so within 12 hours your site should know about any updated versions available.
So YES wp will have a pretty good idea who is running what where - they probably even know about the non-wordpress.org hosted plugins (unl;ess again those plugins have their upgrade checks filtered out or redirected)
In file wp-includes/update.php, wordpress passes the following api call
So on most systems, when updates are done, WordPress.Org knows/checks for compatibility the wp version, the php version, the mysql version, the default locale, the no of users and whether multisite is running.
So over time, on average they'd have a pretty good idea who is running what. This is useful info. EG: for a plugin developer to know how many/few users are already on php 5.3 etc etc
You can see some stats here: