How tight should a fixie's chain be?
The simple answer is, as Sheldon brown says, as tight as possible without binding. 
But define binding. The noises you described on your chain in your previous question were symptomatic of binding or of a worn chain. The description you've given there of how you run your chain tension is tighter than I would recommend. 


  To clarify the tensioning - When I replaced the chain I adjusted the tension. There was a very noticeable adjustment required. But then when I subsequently adjusted the rear cog I didn't adjust the tug from it's setting with the new chain. I agree that I might have changed the tension marginally just be removing and refitting the wheel.


1 millimeter of movement is too tight, in every case I've seen. The manufacturing tolerances on the components (cog, chain, and chain ring, ans well as the hub, and BB axle), would need absolute perfection to allow that tight of tension to function without damage or noise. I've never seen good enough all at once, and you would need to be amazingly lucky or diligent to find that mix of perfection.

Typical tension on a fixie or single speed is roughly 1/4" to 1/2", or 5-10 millimeters, when pressing on the chain from the top, and from one side. A single speed requires slightly less tension, because the bearings in the freewheel will bind, and it won't roll at all, where a fixie will just be noisy and prone to wear. Make sure you check for the spot where the chain is tightest, and do the adjustment there, as well.

You don't need to be that tight, and your drive train will still maintain good contact between pedal and wheel.