I know this is an old post, but just incase people come across this question again, I'll share some of my experience/research for integrating Social Media in Unity so far.
Facebook SDK
Platforms: Android and iOS
For anyone wanting to integrate Facebook login/logout, sharing, and invite functionality - or any of those combinations, I would highly recommend the Facebook SDK which is free on the Unity asset store:
Facebook SDK for Unity (beta)
I used this video tutorial to help get started with Facebook Unity integration:
Facebook SDK 7.4 in Unity 5.3
To share a score you can use the ShareLink function and get the score variable and post that in the description or title for ShareLink.
Let's Tweet In Unity
Platforms: Android and iOS
I tried to work with the Let's Tweet In Unity plugin, but so far I am having difficulties posting a status update in Twitter. I can get access to the account and verified it for Twitter developer, but no luck yet. It is free though and sounds like it would work well once it has been set up properly:
Let's Tweet In Unity Plugin
NOTE: With this Twitter plugin to work with Unity v5+, you have to change the HashTables to Dictionary - so instead of Hashtable, you would have Dictionary < string, string >
To share a score here again accessing the score variable, you should be able to pass in the score as the text parameter for the PostTweet function - although, I am having trouble getting this function to work for me, I assume it would be that simple.
Unishare
Platforms: Android and iOS
There is also the UniShare plugin, but this one costs $7.99. But seems able to share text as well as in game screenshots - and shares to any sharing media type on mobile:
Unishare Unity Plugin
I am unsure how the code works for this plugin because I have not bought it, but I assume there is a simple way to add a standard text message including your score to the share link.