In the interest of clarity, let's look at this in more detail.
Even a shallow 1-2" live aragonite sand bed will perform nitrification and denitrification in a typical reef aquarium. If you were to blow up a single live sand grain to the size of a small country, you'd basically see that the exterior is coated with nitrifiers (they'd have access to high oxygenation) and the interior with denitrifiers (anoxic or low oxygen conditions). The bacteria types are 'coupled', meaning they are in close proximity to each other and can efficiently process the products associated with the nitrogen cycle. You also get stratification of the aerobic, anoxic and anaerobic layers when the sand is not too shallow (~1" -3" with typical medium flow), but much more is doable with a substantial population of sand stirring infauna, but has other risks that have contributed to making deep sand beds less popular (mainly, depleted oxygen during power outages and unsightly appearance).
The area of a typical reef aquarium sand bed is substantial when you consider each grain's surface area (typically much more surface area for nitrification/denitrification than the live rock). One of live rock's 'claim to fames' is that along with some nitrification/denitrification it hosts bio films/periphyton that efficiently remove detritus particles, algae spores, bacteria/archaea/fungi, etc. which in turn provides water clarity (among other benefits).
While benthic organisms do move sand grains around (aka 'bioturbation'), the main movement associated with substances entering the substrate is from a physical process called 'advection' and is brought about by water flow. A secondary process is 'diffusion', but it is typically orders of magnitude less effective in a reef system.
So the short answer is, 'Yes', live aragonite sand can nitrify and denitrify quite well