Sometimes I do, maybe while stretching my body near the toilets or as the airplane is landing at the destination.
Once, I had a long, long conversation with an American woman on a flight from Boston to L.A. because my mother had recently died and her mother was dieing in hospital. When the plane arrived at L.A. we hugged eachother goodbye!
But that's rare.
This summer I am going to the UK and I am going to try KLM's Meet and Seat system.
Before the flight you can see the passenger profiles on Facebook and other social network sites - other passengers who are on your flight!
If they are interesting to you, then you can contact them. Maybe arrange to meet at the airport, or sit next to each other on the plane etc etc.
I will do this on the flight from Amsterdam to Osaka. At the moment I am the only passenger in Economy class who has joined this system...but every week I check the KLM website to see if any other passengers want to Meet and Seat.
Here is a Japanese story about the system.
What do you think? Scary or interesting???