Thank you for your post.
Do you also have FileMaker Pro 9 installed on this machine? If not, install FileMaker Pro 9 and see if you run into the same problem. If the problem exists, then you may have a configuration issue. If it works with FileMaker Pro 9, then create a new User account, log onto the new account, launch FileMaker Pro 10 and see if you are able to access the files remotely. If this works, then you will need to determine what is different about the two user accounts.
If your old machine and new machine are on the same network, try creating a new file on one machine and see if the other machine sees it. Also, try installing FileMaker Pro 10 on your old machine and see if you are able to access the file remotely.
Please continue to update me with any progress.