Does Chrome Remote Desktop work through LAN aswell?

The same question was asked on the Google forums, with an employee responding that the connection is over LAN (except in a few rare cases), but the authentication is done through Google servers.

His full reply:

To be clear, connections between your client and host are direct peer-to-peer. Although the connection setup is mediated by Google servers, once it's established Google is no longer involved (except in very rare cases where we need to fall back and use a relay server). So in cases where your client and host are on the same network, once the connection is made it's likely to perform better than one that has to traverse the public Internet.

Chrome Remote Desktop uses an entirely different kind of protocol from RDP. It uses a video feed instead of sending bitmaps or images, and includes other optimizations to make the experience much better than RDP in most instances.

I have had the experience of using Remote Desktop over LAN. But it needs internet for the initial authentication via the Google servers. After that it's a direct peer-to-peer connection. If the internet goes out, the connection is smartly shifted to LAN if possible.

It happened in my case: the connection was set up and working and then the internet went out. But, instead of the connection getting dropped, it changed its path to LAN and it became faster than ever (Cheers to that!).