Q:

What controls the visibility of the "Share your answer" section? 

As the title states, I've noticed that after answering a question I can share my answer on Facebook/Twitter, yet this option disappears when looking at answers I've posted previously elsewhere. 
Is there a way to access these links when revisiting a question?  

A:

I believe Jeff has said the popup is random.   You can get it anytime you like using the "link" link just under the tags, next to edit, close, flag, etc. 
Edit:
Or did you mean this one: Please ditch all of the share icons - they give me nightmares?   If so, see that post for Jeff's answer. 