If you don't know what Hamachi is, it's a program that created a virtual LAN network for people over the internet to connect to and play as if they were on their own real LAN network.

Anyway, we're on our own network, and try to connect via LAN, to no avail. We can't see each other in the lobby or each other's own games.

Both versions are patched to v1.03. Ports are forwarded.

What can we do?

If you are on Hamachi, you are not really on the same "copper" network, you are on the same IP network, and you are at both sides of a virtual router.
Chessmaster based its LAN communication on NetBios or IPX, and these protocols can't cross network routers.
Well file sharing is also based on netbios on TCP/IP, and Microsoft made them cross routers with wins servers, etc... but that's an other story...