EXPERT RESPONSE
This is one of the best posed questions I have ever been asked. Unfortunately, the answer could be debated for years, just as the same question continues to be debated with regard to house construction, which one would think had been resolved years ago. There is even a question as to whether a vapor barrier is needed at all for an interior data center, but since most people call for it, I can at least tell you what we recommend and why, as well as give you a little of the "debate" thinking.
First of all, the purpose of a vapor barrier is to keep condensation out of insulation, since moisture reduces the insulating value, and to keep it from ultimately rotting the insides of walls. (If the walls don't go from floor to the slab above, this is not a concern since moisture can escape out of the open walls at the top, but a data center wall should be slab-to-slab, both to maintain the interior environment and to provide the fire rating.) The generally accepted rule is that the vapor barrier goes on the warmer side of the wall. The reasons are two-fold. First, warmer air migrates toward colder air, and one would hope that the offices would not normally reach the temperature of the data center; second, warm air is inherently capable of holding more moisture than cold air, so the normal assumption is that any condensation will take place on the warmer side of the wall as the more moist air migrates to the colder side. The goal, therefore, is to stop the moisture before it gets into the insulation.
Assuming you are in the North, in summer the building air conditioning will remove moisture so humidity will be relatively low. In winter, heating will increase air temperature, but the outside air will be crisp and very low in humidity. If you are in the South, you will have high heat and humidity outside in summer, which the air conditioning will reduce, but not as much as in the North. In winter you may still use air conditioning in regions where it never gets really cold, which means lower humidity as well. So to start with, no one analysis holds completely true for every part of the country, but we can still draw a general rule. For an interior data center such as you describe, the "office" or "exterior" side of the wall should still be "warmer" than the data center side.
However, this is where things get a little sticky. In a data center, we are not only cooling to levels below the normal office environment. We are also forcing the humidity to abnormal levels relative to the temperatures we are running. Therefore, it is a virtual certainty that the data center will be both colder and more humid than the offices, at least while building environmental systems (heating or air conditioning) are operating in the office areas. This is not in line with "norms." So even though we would usually think of warmer air as also being more humid, this doesn't hold true between offices and the data center, which throws a bit of a monkey wrench into conventional theory -- except for one thing: While we specify both a relatively low temperature and a relatively high humidity range for the data center, we always tack on the words "non-condensing." Obviously, we don't want a situation where moisture starts condensing on cabinets or equipment, so this phrase makes sure the air coming out of the CRAC units is not made so cold in order to cool the equipment that it is below the dew point when humidity is added. And if it won't condense on our equipment surfaces, it certainly won't condense on or in the interior wall. Therefore, the "outside" wall is still the place where any concern might arise. But let's recognize that this condition is likely to arise only on summer weekends or holidays when building systems are shut down and the office air truly becomes both hot and more humid than that in the data center.
As to the best type of vapor barrier to use, as can be seen from the above description, the barrier itself isn't really going to do much of anything most of the time, so this is not a major concern. Contractors love poly sheeting, but it is more easily damaged and compromised during construction than most any of the other materials. You mention that your insulation is for noise-control purposes, so foil-faced insulation is probably not a good choice (unless it's rigid board insulation) since standard batt insulations do not have the density to provide much sound attenuation. But thermal control inside the data center is usually the reason for insulating the wall, not sound attenuation, since it is expensive to maintain a good data center environment and thick, loose batts are the right choice for that purpose. If you also want noise control, it would probably be best to use thermal insulation between the studs, put a layer of rigid, noise control fiberglass under the gyp board on the data center side of the wall and use foil-faced gyp board on the office side. But frankly, for the amount of condensation control you are actually dealing with, the specific type of vapor barrier is not very important. Whatever your contractor is most comfortable installing is probably the best choice.
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