I am having problems connecting to the Minecraft server from a 2nd machine in the local network

On Service Pack 3, there seems to be a lot of problems you apparently can't get around.

Quoting this topic: http://www.minecraftforum.net/viewtopic.php?f=1013&t=22790#p412298

[INFO] Player1 lost connection: End of stream
Researching
Problem:
  You're running Windows XP Service Pack 3.
Solution:
  Run Minecraft on a Virtual Machine on your computer,
  or use one with a different OS. Sorry!

Yo, I was having the same problem. I found a way. The problem was caused by Atheros L1 network adapter "number of receive buffers" count. The default is 256; after I increased it to 1024, the problem disappeared. This picture shows you how.

Update your network adapter driver on the host computer. Worked for me. I'm going to have a stab and also say that you are using XP SP3 on the host computer. Common problem, apparently...