Facebook share button not showing up
I was finally able to resolve the problem. Just posting the solution in case it comes useful to anyone else. 

As it turned out, the problem with Facebook share was display suite. The nodes that it was not working for were, of course, not random as it initially looked. They were the nodes which were using display suite. I didn't manage to find a solution to it and therefore I used the service links module (proposed by @Kenny Wyland), which had an extra feature covering the case that display suite is being used.