It may be a bit late to add to this thread, but there are a few things that perhaps should be said.
PCB stackup:
the original suggestion was the best one: Sig-GND-GND-Pwr. This works best when the Sig-GND is separated by a thin core (0.05mm to 0.1mm but closer is better), and GND-Pwr will have to match this, for reasons of symmetry. This means that for those times when you have to run signals through part of the Pwr layer (or vice-versa) changing layers means changing GND planes, so linking them with vias is much better than the classic (but terrible) putting GND and Pwr on inner layers because now your displacement currents flow on a layer that is a different voltage, so you need decoupling capacitors where the signals change layer.
PSU layout:
Keep it all on the same layer. A mixture of top layer and bottom layer makes things much harder to design well. Don’t route traces under the switching inductor, even shielded inductors will couple into adjacent traces, and the worst signal to route under the inductor is the feedback signal.
Via stitching:
Stitching between the two GND planes is mandatory, in addition to the comments made by others, this is what prevents cavity resonances, but there are places where doing this is not right.
In the switcher itself, it is best to try to route the GND on the same layer as the other main switcher traces (I presume this will be the top layer). This GND area has to be stitched into the two planes, but when the switcher includes the power MOSFETs internally, your choice is already made because most switchers expect the large GND pad under their bodies to be connected through the PCB to help get rid of heat. In this case, adding more stitching vias in different parts of the GND does not help, but makes system noise worse.
So Stitch the GNDs together across the board generally. But don’t stitch through the area where the switchers are, except for the point (usually under the switcher IC) where you link the GND routed as very wide traces on the top layer into the planes.
There was mention in the initial post of whether to put stitching vias in void areas. The answer is yes, and the reason is cavity resonance avoidance. The associated question was whether to add GND copper to make it worth stitching through all four layers. The answer to this is no, they are more likely to cause trouble, and don’t really give you anything much in return. If you’ve got no need to add this extra copper, leave it off; but keep the stitching vias for the reason already given.
Hope that helps.