[FrankenMauser]

It is possible with many SxS shotguns, but not all.
Some have inertia safeties that, when operating correctly, block the other firing mechanism under recoil.

Manufacturers are aware of the fact that people want light, crisp triggers; but also the fact that people don't care for both chambers firing at the same time.

The inertia safeties were a product of the days when guns were slim, light, and properly balanced, 10 ga was normal, 12 ga was for dove and pigeon sized game, 16 ga was for little birds, 20 ga was for sissies, and a common load was something like 1-3/8 oz of shot on top of a *minimum* of 90 gr of BP.

I have read many reports from the 1870s and 1880s, in which the writers claim to have managed 1-3/4 oz of shot and up to 140 gr of 3F in a 12 ga shell. (Probably brass - but it is almost never specified.)

...Which is why there is so much 'dead' space in modern shells, and cushioning in modern wads. All of our shotgun gauges were built for heavy payloads and large charges of black powder, except .410 bore. .410, as we know it today, was introduced as a smokeless shell. Its predecessors, the various .44 shot cartridges and the ~1901 ".410" were BP; but .44 shot was pretty much gone by 1930, and the early ".410" almost instantly disappears from advertisements, correspondence, and articles after the introduction of the .410 that we have today.
Which I find strange and fascinating. En masse, people adopted a longer, clunkier, more expensive 'garden gauge' (which was given the 2.5" length purely to look more powerful), and abandoned a whole family of compact, effective cartridges that were already in use.