A second interesting observation is that the viewscreen potentially presents a 180 degree FOV's worth of information. Even though this is not visible to someone looking right at the screen from a distance, if you put your cheek right up against the "glass" of the screen you can seemingly see the very corners of the other side's bridge.
Below: The blue door, and the control panel with two levers is only visible from the viewer's POV, not the captain's.
I think it is obvious from that that at the bare minimum, there is a camera(s) on the other end which has a full hemisphere of viewfield, possibly a full sphere.
The easiest way of inmplementing the screen, then, is to build a bowl-shaped alcove or recess into the wall of the bridge, and have the (curved) screen lie along this surface. That could also produce the illusion of the screen "always facing you", using an old low-tech trick:
Given ST's tech level, I would expect there to be an array of scanners and sensors all around the bridge, able to provide a view from any angle at will, and even compose a 3D representation of the scene.
On the receiving end, you could use the same technology as in holodecks to reconstruct the scene in a very lifelike matter.
One thing that always concerned me is the security implications of showing foreign military personnel an image of your starship's bridge. Especially with ST's tech, it shouldn't be hard to edit the video stream online and insert a generic bridge background scene behind the captain. Perhaps this is already happening in the usual case: You could imagine that the comms equipment defaults to transmitting the of the speaker, and the receiving equipment automatically fills in the missing background with a standard scene.