First, there are a number of control and monitoring processes to which you can use a UC, Arduino to monitor or control different variables, but my advice would first be to define the system to which you want to apply the monitoring or actuators you need to control, and then without using the internet you would have many options, wired communication between the Arduinos, I2C, SPI, RS232 or by radio frequency, LORA, SIGFOX, Bluetooth etc ...
Communication between devices, nodes and sinks, the paradigm that best suits your application.
You can review some processes in which we have used an Arduino to monitor some interesting processes below.
Conference Paper System of monitoring and automation of a rural pumping syste...
Conference Paper Temperature monitoring system for the pasteurization process...
If you need any other advice, do not hesitate to ask.
You can also use GSM/GPRS modules, such as the SIM800L which uses the SPI interface. This means full wireless, addressable devices and lower power in receive mode. The only requirement is that the controlled devices must have cellular coverage.
There are plenty good amateur projects in field of Smart House described on Web. For example (need some translation for comments but not for ideas and code ;))) )
https://habr.com/ru/post/382177/
https://habr.com/ru/post/411141/
https://mysku.ru/blog/diy/69609.html
etc.
So you must formulate the terms of reference for your project. And then choose the appropriate development and modify it for your terms of reference.
You can use eithe wired or wireless connections. I would suggest setting up a local intranet server (not internet) on a raspberry pi or laptop for monitoring arduinos. This connection can be either wired using ethernet and router or wirelessly using esp8266 or other wifi shields with router.
If you don't want to use web protocols, another suggestion would be I2C communication/zigbee/bluetooth
To build a remote control system using adruino you must use a adequate shield (to connect them with Wi-Fi or Bluetooth ), in attached you will have an example of conception and realization of a remote control system for electrical installations for home automation :