Our doorbell stopped working a while ago and my wife finally bugged me enough to fix it.
First, I found a diagram on another site when I googled this. It is wired like the right box:
I started by measuring the voltage coming out of the transformer which is attached to a junction box in the basement. There is a small amount of power coming out of the transformer, which itself appears to be new relative to everything else in the house. Just to be sure, I turned off the breaker and checked the connections inside the junction box. Same results after.
The doorbell ringer is in the kitchen. When I take off the cover and measure the voltage it matches what I see in the basement. However, when I press either doorbell it does not ring: if I short circuit the screws for a doorbell using a wire, nothing happens: unless I am wrong this should make it ring since it is making a complete circuit and essentially removing the switch/button.
Two other possibly relevant pieces of information. First, one of the doorbell buttons is damaged, it is smashed beyond repair: I have to replace this anyway. Second, my wife swears this all started when my brother (an electrician) replaced the load center a while ago. Since the transformer is 30 feet from the load center and everything else on that circuit works fine, I am doubtful that is the cause. So these are probably red herrings but I have to tell my wife I tried.
My educated guess is that the doorbell unit itself is faulty and needs to be replaced.
My question is this: should I check anything else before spending the money to replace it?
UPDATE: I disconnected the wires from the transformer at the doorbell ringer. I connected it to an AC outlet using one of those adjustable wall-warts. It only goes up to 12V (transformer is 16V) but it rang when I shorted it to one of the switch wires. So I think this is the transformer.
Are these transformers fixable (I do hobby electronics and have rewired rooms in my house)? Or should I buy a new one?
UPDATE 2: I just learned there are two black wires coming out of the transformer, and one of them (does not matter which) needs to be wired to neutral. When this broke, I had replaced the ancient junction box the transformer was attached to with one twice as deep and added conduit to go to an outlet in that under-stairs closet. Everything else worked so I did not think much of it. I am still not sure why one of the wires would not be white since that would make sense, but oh well. I just wired it correctly and it works fine. I will accept BMitch's answer since it is close and helpful.
Also, the erratic results on my multimeter were likely due to both wires being hot. I knew something was not right but should have spent time figuring that out instead of looking elsewhere. Oh well, live and and learn. Thank you all for the feedback.