Chime question

Does the Chime device when used on the same wifi network as the Ring Doorbell depend upon the Ring servers in order to produce the doorbell chime when the button is pressed on the Ring doorbell?  If the answer is yes, I would expect delays as I experience with notifications when my Ring Spotlight camera detects motion.  Sometimes those delays are only a couple of seconds.  Sometimes it can be minutes.  Often, the delivery person is gone by the time my wifi-connect phone receives the alert when I am home. I am considering a Ring doorbell, but this could be a deal breaker if the Chime device doesn't sound when the button is pressed.