A simple screen can be a bit dull, so how can it be made fancier? I like the idea of a door, so let's add :

• an actual door that will open on the screen and on the camera. so if the door is closed, the user on the other side can't see you
• an accelerometer on the door, so it can detect if you knock on it, it's nicer than a simple button.
  It could also be used to make a crinking noise when the door is being opened
• an actuator (servo, vibrator) that can knock or "shake" the door a bit to make it looks like there someone else knocking on the other side
• a peephole with lenses and a proximity sensor
• a sensor that detects if the door is open

How it would work
Your door, A is connected to a single other door B
When you knock on your door A

• It makes the connected door B shake  
• once someone has opened the door B, your door A signals it can be opened.
  In a first time, the signaling will be done with a sound or a light, but in a second time, we can imagine the door is locked and will only unlock if someone else open on the other side.
• the video starts streaming
  if nobody is on the other side, the "door" will ask you to try later and if you open it the screen won't display any video stream from the other side

Using the peephole
I like the idea of a peephole but I'm not sure how to make it interesting without being too invading for the privacy.
It will probably display a static image or a looping video of the other side, a bit like a profile picture, so you can know what to expect before you knock the door.
I don't know how to display the image, I can either add a 1" OLED screen inside the door, of I can just make a real hole in the door, add lenses, and a sensor so that when someone takes a peek, it switches on the screen and display an image on the part which is just in front of the peephole, this way of doing is a bit more hackerish ;-)

Half skype, half chatroulette
In the original concept case the connectivity is simple a door A is always linked to a door B. But we can imagine a more dynamic connectivity where, when a user knock on a door a requests is sent on a server to find a peer. Or you can make other rules like "a door opened in a museum will always open on a door located in another museum, a door in your hackerspace will always open on a door in another hackerspace" etc.
You could also put restrictions on the language used, and for example only request Spanish speaking door if the door is located in your Spanish language exchange club, etc.

But honestly this feature would only make sense if hundreds of "doors" are in use, or if there is a software version, let's focus on something more realistic first ;-)

Illustration