OK, this may be a side-effect of the way the Ring powers itself.
The Ring is in series with the chime. In series circuits, the same amount of current (amps) flows around the loop through both devices. And by the way, this problem crops up with Nest thermostats and dimmers too.
How does the Ring power itself? By deliberately letting a little bit of current flow through the circuit, including through the chime. The Ring (like the Nest and dimmers) is betting that a small enough current won't make the chime (furnace, lamp) activate. And it is guessing at how much current it can get away with.
So my guess is: you have a chime which is more efficient than most. Sure, when the visitor rings the bell, the Ring shorts the two wires to simulate a doorbell push, and then it lets go of that correctly. And then, it needs to recharge its internal battery - transmitting video over WiFi is hard work. So it allows current to flow at that lower rate. Unbeknownst to the Ring, that is enough to ring the chime. The Ring isn't timing out when it stops; it's just finished charging.
Now like I say, smart thermostats have the same problem: they charge by sending some current down the W line, hopefully not enough to engage the furnace relay. If that doesn't work, they give you an option to hook up the "C" line to power the thermostat directly. Unfortunately it does not appear the Ring has an option like that.
TLDR: Your chime is too efficient for the Ring to vampire-charge itself. Get rid of one.
Well, if you're in love with that chime, there might be another option, but it's definitely not UL-approved. You could alter your chime to be less efficient... by putting a resistor of appropriate size in parallel with the chime. Finding the appropriate size would take some measuring or experimentation. You would not want one which gets too hot, worst case someone holds down your doorbell button for a long time. A fellow shouldn't be able to burn your house down by holding the doorbell button.
Most likely, the doorbell button has failed.
There will always be voltage across the doorbell button unless it is pushed. That's because the circuit would be complete, but for the button. I would expect to see 18 volts (or bare transformer voltage), but with an electronic chime, I could see that value being a little funny.
You can test this out; short the 2 wires to the doorbell, that is the same as ringing the doorbell.
Best Answer
Generally the chime will have a solenoid that gets pulled or pushed when the circuit is completed. The solenoid bangs against the metal and you hear the nice chime. [I thought these were 24V circuits - so be sure you check your door bell system out and verify that first.
Check your voltage on the solenoid - you can check resistance with no power attached (should be some low ohms around 4 ohms probably.) When your door bell button is pushed voltage will flow from the Transformer into your solenoid - so checking the voltage at the solenoid will provide an answer as to what is working and what is not working.
You should be able to measure your 'voltage drop' at the solenoid when the doorbell circuit is made. Use Ohms Law - the voltage you know is 16V, the ohms is the resistance of the solenoid. Your solenoid even though it is an inductor has resistance - so it is your equivalent to a resistor in the example link.
Here is a quick link: http://www.wikihow.com/Calculate-Voltage-Across-a-Resistor
Some things to keep in mind: 1: Solenoid sticking - clean it and use graphite lock type lubricant. 2: Solenoid open or shorted.. 3: Transformer not working or "Overloaded" - too much current draw causes the voltage on the transformer to drop and it will not be able to pull in the solenoid. This indicates either something shorted - or a wrong doorbell chime, or a transformer that is simply worn out. 4: Doorbell switch bad .. or too much resistance - simply put one wire to the other and see if the thing works - if it does the switch is bad.