Thank you so much for the wonderful comment that prompted this post. The question was posed in terms of Waterbury and Irish immigrants, but it could also apply to Italian or French-Canadian immigrants: how do you locate the Catholic church that might hold their records? The ethnic church of today might have looked very different when your ancestor arrived.
I'd be the first to admit my technique isn't perfect, but it works fairly well in Middlesex County, where I am most comfortable. I start by determining the current list of Catholic churches in the town using Google. I then open each church's website. Most churches - although not all - have a web page with a short history of the church. Look for and record each church's establishment date and ethnic traditions. Soon you will have a timeline that you can compare against major events in your ancestor's life. If there are two or more churches in the area, first check the church closest to your ancestor's residence. Second, consider using your list of ethnic traditions. Your ancestor could have attended either one. Good luck!