As per above I don't see why the person you're sharing a desk with has to unplug.
On the following schematics I've used --> <-- to represent a male/female connector. 1/2 represent ports on the respective docks, whilst A/B represent connections to individual monitors.
Correct if wrong but at the moment you have...
--------------------
| Your dock |---> 1 <---------------------> A <--------- Monitor 1
-------------------- |
------------------> B <--------- Monitor 2
And this guy is coming in and doing....
--------------------
| His dock |---> 1 (A) <------------------------------ Monitor 1
--------------------
|--------> 2 (B) <------------------------------ Monitor 2
So basically he's removing the splitter that you have to use to get the signal going to both monitors ('cause his dock has this built in).
Instead of him doing this I don't see why he can't unplug 1 on your dock and plug it into 1 on his dock and use the splitter that you do to get his output onto two monitors.
Just tell him this (or perhaps ask if it works and if it does ask him to use this method of connecting the dual-head to his dock than his current method of unplugging your splitter).
Certainly a cheaper option the switches!
Although I may have misunderstood something crucial, won't be the first nor last time