Player1 starts a single player game on Computer1, then hits "Open To Lan". Player2 joins on Computer2. Player2 ends up with an exact duplicate of Player1's inventory and position. That's the first problem.

Then, both players go off and play and accumulate distinct inventories. Player1 ends the game (disconnecting Player2). Then:

Player2 logs into his Mojang account on Computer1 and starts playing. Player2 gets Player1's inventory and position! The [username].dat file appears to get overwritten and everything.

Have tested this on 1.7.4 and 1.6.4, with the latest launcher.