I have an alternative (but probably equally as effective) solution. I was a little nervous about just randomly sticking a wire in there without knowing more about what was going on, so I googled a little more about the problem. People have the same issue with Ring Pro, August, etc. August has a solution called the "August Chime Power Board" that is listed on their store for $0.00 with free shipping, so I ordered it. I figured I would get contacted by them to confirm my ownership of an August doorbell, but I got a shipping notification the next day. I got the power board today and it was a 2 minute install and the buzzing is completely gone and my RCA doorbell still works great. Before I installed it I took it apart, it's a very simple board. Here are some pictures:Good news. I think I may have a solution to the buzzing in the chime. At least it worked for me. Most chimes should have 3 terminals. One is common, in the middle. The other 2 are for the front door and the back door. Just bridge the front and back door terminals with a wire.
I have a picture and an explanation on my blog post at the end. Please let me know if this worked for you.
Installing the Nelly Security WiFi Video Doorbell
Maybe someone who knows a little bit more about circuits can tell me what exactly is in this board. To me it just looks like a resistor and some sort of IC. I'm wondering what this does opposed to the solution that @theneedfull came up with.
I would encourage everyone to refrain from taking advantage of August's "free" solution. I feel kind of bad for even tipping everyone off to this. Hopefully we can come up with a simple DIY substitute, or someone can confirm why @theneedfull's solution works and why it's safe