It seems that enabling Page Cache prevents social sharing.
When enabled, articles can't be submitted to Digg or StumbleUpon.
The 4 options I had enabled for it were:
-Cache home page
-Cache SSL (https) requests
-Cache requests only for hostname
-Don't cache pages for logged in users
I've tried enabling/disabling all of them, but even all disabled would still prevent sharing. Only by disabling Page Cache under General was sharing possible.
Any ideas on why this could be?