Short Version:
Usually the modem/router you use to connect to your home computers to the internet will control what gets into your network.
For an external entity to gain access to your internal WordPress installation you would generally need port 80 forwarded to your WebServer/WordPress install. This is something that can only happen if you (or someone else) explicitly allows. Without that, the world at large won't be accessing the install.
By Installing the basic tools of LAMP and WordPress and no other "features", you will not be sharing your install externally, and it will be safe.
That's not to say you shouldn't take extra precautions (for example, attacks from within your network), but largely you are covered.
Longer Version:
There are a number of ways that an external person CAN connect to an internal WebServer, but they all involve explicitly creating an entry point, for example:
- (as mentioned) forwarding of port 80 (or other) from the outside world into your network (this would allow the world at large to connect to your WordPress install),
- a VPN setup (which would allow only the authorised users access to the internal network).
- a SSH tunnel (again, very explicit users), or
- an entry point created via malicious means (malicious email attachment, malware file, etc)
If you and your partner are operating on your own secure network and already have a connection method between computers in your network (NAT within the router) then setting up WordPress as an internal CMS is very easy to do with several great resources out there to achieve that (I know you said you're probably fine with the installation, but I will share one great resource for installing WordPress on Ubuntu, and that is Digital Ocean's:
About the only other consideration would be if you were installing it on a Windows machine, or a machine with an overly aggressive software/OS firewall, but you've mentioned Ubuntu and this will allow the share for you (once setup correctly).