I'm just noticing this thread. Sounds like the door bell is just the old style with a couple solenoids. you press the front door bell button and the solenoid's armature strikes the first chime(Ding). Release the button, and the spring sends it back to hit the other chime(Dong). The back door bell button causes the 2nd solenoid's armature to hit he 1st chime(Ding), but the spring return doesn't cause it to hit anything.
A typical button is open or closed, or perhaps there is an very hi resistance light in the button. No big deal. But if your button becomes a parasitic load, it's flowing current all the time. Maybe enough to cause buzzing. So you push the button, you cause max current to flow and you strike the first bar(Ding). Then, you release the button and the parasitic load still allows enough current flow to fight against the spring return. (No Dong) So in a case like that, you would need to use an interface between the old and the new.