The Ring Video Doorbell 2, uses power from the existing chime transformer to charge the Ring battery and power the existing mechanical chime. The most common transformer is 16 volt 10VA. VA is volts x amps, similar to a measurement in watts. VA is a measurement of Power.
If your transformer is 10VA, it may be under powered. It may not be powerful enough to charge the battery and run the Video. Change it to 16 volt, 15VA or even better 16 volt, 30VA.
When I first installed mine, I had a 10VA transformer. Both the Ring Video Doorbell 2 and Chime worked, but the Ring battery would slowly drain and the chime sound was weak. I changed to 30VA, now the Ring is always 98-100% charged and the Chime is strong.