Connecting two phone handsets to talk
Your microphone will need to have a simple amplifier in order to power the speaker. This might depend a little bit on what speaker is actually being used in the headset but an op-amp circuit should work for you. You will just have to make sure that the op-amp can output enough power to power the headset. If standard op-amps are not powerful enough to power to speaker you will need to go with a more complicated amplifier.

As for the toggling the connection, are you wanting to just have a dedicated 1-1 connection that can either be on or off? If so, you could do this by powering off the amps with a simple switch. If you are wanting some switch capabilities, such as 1 phone can be routed to 2 different locations, you should look into analog multiplexers.