Access a MacMini in a private network though Internet

Yes you can setup the Mac mini for File and Screen Sharing over your local Wi-Fi network.

To setup File and Screen Sharing, you will need to connect a Display, Keyboard and Mouse to the Mac mini.

see > Set up file sharing on Mac - Apple Support

and > Turn Mac screen sharing on or off - Apple Support

Also please provide more details about the Mac mini, the other computer(s) and exactly what you want to do.

see > Writing an effective Apple Support Commun… - Apple Community

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