I want to install a doorbell with two bells and one button so it can be heard in two places. One location is within 25ft of the button. The other is about 200ft from the button. Will the transformer work from 200ft away or should I place it so it's 100ft from each bell?
Not a standard door bell transformer no. #1 A standard door bell transformer is sized to drive one chime only. If you want a second chime you need to design a system. I recommend you call a manufacturer of chime kits and ask for a tech rep. Broan, Nutone. Tell him your situation. He will tell you what you need. A standard chime transformer are 16-volts. I have used 24-volt transformers on longer runs with success.
You're better off dealing with the long run as a separate issue, rather than making your long runs even longer. Remember the power has to come to the button first and then split to the 2 bells, so moving the transformer halfway between means you now have a 300ft run. Put the transformer near the button.
Heed Paul Logan's advice about overloading the transformer. But you need to do a voltage drop calculation on the long run. For that you need to know the current drawn by the bell - the maker will know.
Pick a candidate wire size, e.g. 18 AWG. Look up that size wire resistance, 6.385 ohms per 1000 ft. Your round trip is 400 ft, so
6.385 x 400 / 1000 = 2.554 ohms per 400 feet.
Now you have your current, I, and you plug it in to Ohm's Law.
E (voltage drop) = I R
E = your I x 2.554
Now compare your voltage drop E with your transformer voltage. If drop is more than about 10%, that's a big problem. Go redo the calculations with a larger wire size.
You should know which wire sizes are unusually cheap because they ae made in mass quantity. 18AWG thermostat cable is widely used and comes in 2-conductor. 16AWG cordage might be a value, but it's not legal for permanent building wiring. 14AWG in /2 cable is cheapest for its conductivity because it's massively produced for building wiring.
If that isn't big enough, it's time to look for a new approach, like a relay that draws a tiny amount of current and switches power from a local transformer. The relay method would also solve Paul Logan's concerns about overloading the transformer.