Over a decade ago, when I first started building websites in earnest, backlinks were all the rage. What you wanted was to contact other, similar websites in order to get that site to link to you. In turn, you'd link back. Some services bypassed the spirit of this, creating link farms that linked all kinds of related and unrelated websites together. These links, I am sure, included the farm's own link in the mix. Think about the math and how those sites could be rated as having thousands of links without having any real content. But that's the goal, right? Since content is so hard to create, we need to fool Google into thinking our websites are more important than they are.
Somewhere along the line, Google figured out how to avoid linking to pages that had only the merit of other links, and people started abandoning the process. But it still matters. If someone writes a relevant article and links to your website, it certainly matters. If you link to your district office, it matters, and vice versa. You just have to be sure you're not linking out to every website under the sun in order to get meaningless links in return.
While I could see local churches competing, it's probably a good idea to link to similar churches a few counties or states away. Why not, especially if you know that church is very similar. If your church gets a link from one church in each state, that's 49 outside links from similar websites. Not only would you become the recommended church for folks on vacation from the other states, but you're also going to see an increase in overall traffic.
The problem is that if it's just a generic link to your homepage, it's probably not as effective. And neither is 50 links to an article about a Cub Scout meeting from 2014. Write one insightful article, ask for links, and then offer to link back to an insightful article from the church that is linking to you. If you say something about it first, even better. An example of this might be, "I read this article from St. Paul's in Blankville about blank and blank. It was very interesting, and I thought you might enjoy it. Here's the link." Simple. You have one more article to drive traffic to your site, and you're linking out to someone who has helped you by linking in. It's not SEO magic, but it does work.
The biggest challenge is really making contact with the other churches. I have had almost no luck doing this via email because of all the spam and scams that exist. You might need a group of volunteers to help send out a message to other churches, since it seems to be a 1 in 10 ratio of actually getting some kind of response, even when you're just making a helpful suggestion.