If you do not want to enable remote access, then you need to network your two servers together using a VPN solution of some kind. You mention using wireguard, and that should work. However, you need to able to find eachother.
Technical description, skip at will:
The way Plex works, is your clients do not simply “scan” for a local server, they need to be told the IP address of the server itself, even in the same network. The reason this works currently is because your server knows its own private IP address, and your house’s public IP address, and sends both of these IPs to Plex (the company) automatically. When you log in as your client, you receive both IP addresses as potential locations to reach the server. So, let’s say your server is at IP 192.168.0.10, and your public IP is 15.15.15.15. Your client in your house will attempt to reach 192.168.0.10, finds your server, and connects right away. If you are out of the house, your phone would try 192.168.0.10, find nothing (because you are on cellular), then it would try 15.15.15.15, and (assuming you port forwarded) find the server.
End technical description
When you use wireguard, you likely have a new “special” IP address range for all the computers/servers at your friend’s house. You need to find this IP address, and your friend will need to place the IP of his server FROM YOUR PERSPECTIVE into his server settings, in the Settings > Network > Custom server access URLs
box. You need to do the same on your server, but for your friend’s perspective.
It’s possible that it keeps the last digit the same, but you are probably getting a subnet change with wireguard. So from your friend’s perspective, YOUR network might appear to be 192.168.10.X, and your server can be accessed at 192.168.10.10. So you would put 192.168.10.10 in the box I mentioned up above (EDIT: into your friend’s box, not yours). YOUR IP IS LIKELY DIFFERENT, DON’T JUST ENTER THAT IP WITHOUT CHECKING FOR YOURSELF
I would expect wireguard settings in your router should state what the IP range of your friend’s network is, and you use that to find out.