Is it possible to edit files on a remote machine using the MATE command to work on a project / directory locally?
If you have FTP/SFTP access to the machine use Transmit to access that machine.
In Transmit's "Files" preferences pane set TextMate as the default "Custom File Editor".
Use Transmit's file browser to find the file you want then you can use the "Edit in Textmate" from the right click menu or from File->Edit with External Editor (Command-J). Now when you save in Textmate it will send the saved file to the remote machine.
I believe that other FTP clients have the same or similar functionality, but Transmit is only $30.
Good Luck.
In addition to @Mark's suggestion, you can use Cyberduck instead of Transmit.
It's free and allows remote files to be edited using TextMate.
The solution is to mount the remote system using SSHFS, on OSX the best option for this is MacFUSE.
The filesystem is mounted via SSH using your regular SSH credentials or key so it's very easy to do, and then you navigate it as if it's a local directory and enjoy the benefit of having access to all your local environment's resources.