There are three possible solutions to this problem to create a better user experience that also makes it easy to track links:
1. Enlist Stack Exchange developers in displaying the number of clicks beside each shared link in a "shared links stats" page.
The argument for this feature is it may encourage more sharing of links, which is especially important on beta sites. The argument against this feature is that there are already existing tools that track clicks on links. If the argument against this feature wins, then see the next proposed solution...
2. Make it easier to use url shortening services by integrating them with the "share" links below each post.
Many of these services have an API. Stack Exchange developers could implement this, but any developer could implement this as a Stack App user script or browser add-on!**
By making it easier to use url shortening services, we encourage link-sharing without making people go through extra steps who wish to track links. While the first option may potentially reach more users, the second option is potentially easier to implement and allows users to consider to use existing tracking tools outside of Stack Exchange.
The share links that you see today under each post, implemented by Stack Exchange, was originally created as a user script on Stack Apps. It may be possible to use that code as a starting point to expand the share links and integrate them with popular url shortening services.
3. Enlist Stack Exchange API developers to add the stats to the Stack Exchange API.
Adding the stats to the Stack Exchange API seems like a good first step, since the data can at least be made available to add-on/extension and user script developers while not bogging down Stack Exchange developers/designers with more stuff to do.
Option 3 is more advantageous than option 2 because it exposes data for links users have currently shared. For instance, I have links I've been sharing for more than two years. Unless Stack Exchange exposes this information, it's unavailable for any links I haven't shared using shorteners.