The cell phone and ring doorbell are connected to the house WIFI. The cell phone signal is very weak, or non-existent, but WIFI is strong. Will the cell phone make a sound when the Ring doorbell button is hit?
You need an Internet connection. As long as you have a good one, the signal will go out over the internet to your Ring account. You can then access that information (or notifications) via your phone that has an internet connection. If your phone doesn’t have a cell connection, but does have a WiFi connection then you should get notifications at home.