"Note: the app will be running in an environment where there is no Internet. Hence push notification will not work here."

Emphasis original.

The author did not say they have no network connection, they said they have no Internet connection. The two are not synonyms. This makes perfect sense. It is factually wrong to laugh at that.

Now, it is possible that shiplu.mokadd.im is using the terms synonymously, but the text of the question leads me to believe that it is entirely possible that a local network resource is being used over Wifi and that he understands this quite well.

Yes, that part could have been explained a little better, but it seems fairly clear that the person is developing a chat app to run on a local area network. There are a couple ways to do that. One is bluetooth, which has a very limited range. Another is radio signals, which I've never heard of anyone using in an iPhone app (but then again, I don't know much). Another is Wifi direct, which I've heard is very insecure, but again I don't know. It has the advantage of not requiring a central router/server.

Would someone who knows more than me care to flesh out any of these points, since I clearly don't have much information and would like to know more?

The author does specify both that this is not "Internet" (with capital I) and that they are not using bonjour, and that it is a proprietary technology. So it does not appear to be a mistake on the author's part. That is why the suggested solutions, where a third party http/https endpoint is needed, are not appropriate, as there would be no accessibility to those services.

According to the original asker:

"There is no Internet. we are providing chat and other services to people who dont have Internet."

and

"No, we are not using bonjour. Its a proprietary technology. We just provide chat service."