Using WiFi and ethernet networks at the same time

There are gaps in your description that makes it hard to be precise, but I can at least offer some pointers.

First, it's entirely possible to use multiple networks. I do it all the time.

It's clear you have two different network spaces - 192.168.1.x and 10.10.20.x, so that's a good start.

What's missing:

Is the wired connection (192.168.1.x) completely private? There is no connection to the internet over this network?

Is the wireless network your path to the wider internet?

How are you trying to use the Mac Mini? You state "... so that I can connect to it remotely", but using what protocol? screen sharing? SSH? etc?

Additionally, where are you trying to connect from? From another device on the wireless 10.10.20.x network? or from some random device on the wider internet?

In general, whatever connection has your public internet access (e.g. your wireless network) should be listed first in your network services order (i.e. top of the list). There's an option in System Settings to Set Service Order to do this.

In this way, the Mac Mini can access both local networks directly, and any non-local network traffic would be routed across to the router on the wireless network.

To allow incoming access from the internet, you're going to have to setup access controls and/or port forwarding on the wireless router, to permit incoming traffic into the network to get to the Mini.

More than that will require additional details as to your setup and goals.

- I am away from the Mac mini now, but I will ask someone there to check what the order of the networks is and make sure the Wifi is the first one.

Given what you've said, I'm 99% sure that the service order is going to the crux of your problem.

Think of it this way - when the Mac Mini tries to reach a non-local address, it uses the network services in the order listed in System Settings to work out how to get to the Internet, and as you've stated the wired network is entirely private.

You might also get away with making sure there is no router address set on the wired interface - this should force a fallback to the next interface, but setting the service order is a better option.

You'll also need to setup port forwarding on your wifi router to permit incoming traffic to get to the Mac Mini (by default, incoming traffic should be blocked).

> As a plan B, could we connect a wired internet access from the building to the switch I am using for my devices and access the internet with that network?

Conceptually, yes, but there are a lot of details that could derail that, especially since you say:

> wired internet is likely not on the same network space as my private network

Do you mean on a different network space as the 192.168.1.x address space we already covered?

That might work, but connecting two disparate networks to the same switch can have all kinds of problems.

For example, how do devices in each network get their IP addresses? Is everything hard-coded, with someone manually assigning IP addresses to each device? or do you have a DHCP server?

If your 192.168.1.x network has a DHCP server, and the network you want to connect for public internet access ALSO has a DHCP server, you've now connected two DHCP servers to your switch and there is no simple way to control which one responds to any device (in the worst case, your private-only devices could suddenly get a public-facing IP address, which you probably don't want).

There are ways of segregating the switch into 'virtual LANs' (or VLANs) so you can run two separate networks on the same switch, but you first need to check if the switch supports that (not all do, especially low-end consumer switches), and you have to configure each port on the switch to be in LAN 1 or LAN 2 (or both, in the case of the Mac Mini).

So there's lots to unpack there. No way of knowing if it will work without more details.