Games : Local Minecraft server - connect from 2 computers with same account?

I'm running a local minecraft server, and would like to connect to the server from 2 different computers, with the same Minecraft account.

Can I do this? When I try, it says

Failed to login: Invalid session (Try restarting your game)

and kicks the user from the game, so only one user can play at a time. Is there a way to work around this without buying another Minecraft account? I'm willing to try any other minecraft server if need be.