Android : Communication with server without INTERNET permission

I am planning to develop an Android app that communicates with a server that I operate. Answers to android, httpurlconnection error state that classes such as HttpURLConnection and HttpsURLConnection require granting the "full network access" permission (android.permission.INTERNET) to the app. Someone on a forum told me that for the vast majority of apps, android.permission.INTERNET is unacceptably intrusive on the user's privacy, and that there exist other ways for an app to communicate with a server operated by its developer that do not require such an intrusive permission. From this post:

And the same app will ask for full network access, even though if you look at the traffic, almost all of them are using HTTP to talk to their services, and they have no legit need for full network access. However, it lets them look at what all your network connections are.

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Also, no, you don't need "full network access" to access anything outside a web browser. You only need it to go off port 80 HTTP. You can still just use a subdomain for the remote app API. You're conflating two different permissions.

When I asked for further clarification on how to get this going in Android, so that I could go look it up on developer.android.com, the reply was "your ignorance doesn't demonstrate anything."

So how should an app communicate with a server operated by its developer without android.permission.INTERNET? Or is there a reliable source stating that this is impossible in Android?