I saw multiple posts online which said it's reduced by 1 / sqrt(2), but they offered no explanation of why this is the case.

This one is easy to explain. The typical Bayer tile has two identical green-filtered photosites and one instance of each of red- and blue-filtered photosites. The green-filtered ones are usually on the diagonal.

Suppose the horizontal (and vertical) distance between neighbor photosites is a. We have three lattices of identical photosites: two lattices with period 2 a (red and blue, lattice vectors horizontal and vertical in both of these), and one lattice with period a√2 (green, lattice vectors point along diagonals).

Now, suppose we have a camera without any antialiasing filters, and take a photo of a scene in perfect focus. If we are only interested in the green component, we can simply take our green-filtered photosites, rotate the raw image by 45°, and then (after correction for black level, nonlinearity etc.) we get our (rotated) photo with the resolution √2 smaller than that of the sensor itself.

Of course, in real life we are interested in the full color, so we want to use red and green components too. And in real-life images the values of red- and blue-filtered photosites are correlated with the values of the neighboring green-filtered ones. Good demosaicing algorithms can take this into account and yield even better resolution — up to native sensor resolution.

But this improvement is a gamble. You can easily get various artifacts lowering image quality down to 2 times smaller resolution than that of the unfiltered sensor. So in practice effective resolution depends on the scene and is between 1× and 2× the resolution of the unfiltered sensor, with a good guess indeed being 1/√2 × native resolution.