Is it normal if user gets logged in Facebook or Twitter after they use Like or Tweet button?

Personally I don't use social networks, so I'm not sure about their behaviour. Currently I must code website which heavily depends on "Likes" and "Tweets" about articles. Each article has Like and Tweet buttons. We expect that an average user will "Like" or "Tweet" 3-5 articles each day.

1. 
   

   User opens our page, finds an interesting article

   
   


2. 
   

   User clicks on "Like" button

   
   


3. 
   

   Facebook asks for credentials in a popup

   
   


4. 
   

   Popup closes, Likes increment by one

   
   


5. 
   

   However, if user decided to leave our page and visit his Facebook account - it turns out he is logged in. If he navigates around our page and decides to Like more articles - he doesn't need to type his credentials again because FB session is already in place.

   
   

It's fine if user is at home or uses his phone. However, that's a risk for public access computers (school, library). Is there anything I can do to lower the risk? FB session seems to be long-lived. For me it WAS NOT obvious that you will be logged in FB.. Maybe I'm just too old.