Looking at your question I am not sure that it will be possible to do what you want. At least not without first setting up a shared folder unless you plan on connecting to the other computer via another protocol such as FTP (Meaning that you would need an FTP server to be active on that machine).
If you did set up a sharing folder or a network share then you would still need to use the full network address (if it is static, if it is changing then it will make life even more difficult).
Standard read/Write of a file in VB.NET is pretty simple however I have not tested more than one simultaneous operation by more than 1 computer yet... So if anyone knows the answer to that then please let us know lol.
What is the purpose of this piece of code? There may already be a program or otherwise out there already that can help you to solve this problem.
Sorry I cannot be much help but if you need a hand with reading/writing to a txt file in VB.NET then just let me know and I'll send you some code.