The Visual Studio Agent software allows tests to be run on other machines. The Agent software does not need and version of Visual Studio installed on the same machine.
The agent software includes all the DLLs etc needed to run Coded UI tests. So you should be able (but I have not verified this) to load the DLL of your Coded UI test onto the partner machine and then run the test with mstest.exe.
See http://www.microsoft.com/en-gb/download/details.aspx?id=38186
Note that Visual Studio Professional does not include Coded UI test facilities, they are provided in the Premium and Ultimate versions.