Observed Results:

Attempting to join a LAN game without an internet connection causes it to state that it is unable to connect to the authentication servers, whereas solely one user being connected to the internet causes it to state that the session is invalid. Noteworthily, if I use mods when I induce this error, I see a third message:

The authentication are currently down for maintenance.

Regardless of whether using any modifications or not, an internet connection on both client and host is necessary to connect, yet once client and host are connected, the internet connection is unnecessary.

Expected Results:

An Internet connection shouldn't be necessary when connecting via LAN.

Screenshots/Videos:

I should have acquired these. I shall endeavour to.

Extra:

https://launcher.mojang.com/download/Minecraft.tar.gz on https://download.fedoraproject.org/pub/fedora/linux/releases/40/Spins/x86_64/iso/Fedora-KDE-Live-x86_64-40-1.14.iso.
For the sake of versioning, specifically the undermentioned:

flatpak install 'https://dl.flathub.org/repo/appstream/com.mojang.Minecraft.flatpakref' && \
flatpak update 'app/com.mojang.Minecraft/x86_64/stable' --commit=bb7110c933f26a839a8fff4b86b9ff4ed954d1f742f30118d62f995af3441989

However, I've also tested the undermentioned:

flatpak install 'https://dl.flathub.org/repo/appstream/org.prismlauncher.PrismLauncher.flatpakref' && \
flatpak update 'app/org.prismlauncher.PrismLauncher/x86_64/stable' --commit=369388b1bb5a75de1e5181d85810ae42c0302fb344eea63511bfe9326d06f146

...and it occurred there.