Since you have a light, you have a circuit, so the most likely problem is that the solenoid for the rear is stuck. Open the chime and see if both solenoids move freely
The light does go out when pressing the button. It actually is a new button. It hasn't worked in the 4 years we lived here, but I figure it is about time it gets fixed.
Four years is not bad. I recently fixed mine after 20 years with neither the front or back working.
If the switch is OK it is the chime. You can usually just touch the wires together and make the chime work. If it does, that means the switch is bad. If it does not, it means a wire or the chime is bad. You can also try reversing the wires in the chime and see if the rear chime works when the button is pushed on the front.
For my side, I put up a motion detector and set it to come on day or night. Then I spiced a wire onto the hot line to the light and brought it back into the house. Then I put up a timed relay and ran the doorbell circuit through the relay. Now, if anyone walks up my driveway the motion detector goes on and it rings my doorbell. It's pretty slick and people always wonder why I meet them at the rear door.
The doorbell here is a standard chime that's supposed to sound "ding dong" for the front door and just "ding" for the back. When I moved in, those two were reversed. I tried to fix it by reversing he "front" and "back" wires at the chime. Then the front worked right but the back didn't work at all. It turned out the common connection was wired wrong too.
I didn't have any lighted buttons then, and don't know if they still would have lit.
Just Sunday I was at a stranger's house and I wanted to find out what kind of tree was growning in their front yard. No car there so I wrote a note. Went to the carport door to post the note and saw the tv was on in the kitchen. So I range the bell instead.
No answer. So I went around front and rang the bell. Then I worried that the person inside would be running from door to door. So I stood where I could see both doors. :)
The room AC was on too, even though it wasn't hot out.
I left the note, and they called me later, intrastate long distance! It was a sycamore, 50 or 60 feet high, pruned so that it had the shape of an arbor vitae. Except big, 10 feet in diameter, excep for the first 8 feet which were about 4 feet in diameter. I've never seen a tree like it, but I don't have room.
I went to a party once, and when I knocked on the door the dog started barking and the lights inside went on. I waited and knocked again, and waited and knocked, and eventually realized I was at the wrong address. I'm not even sure it was a real dog.
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