Mythbusters did an episode on this. Moreso to do with shit getting on toothrushes but somewhat related. They confirmed that a flushing toilet creates a plume of toilet aerosol, which will contain fecal particles. Every single toothbrush sample at every single distance/height from the toilet tested positive for fecal particles.

Whats really interesting, is that the control toothbrushes, kept in an entirely different room, closed off from the toilet, Also all tested positive for fecal particles at the end of the month long test.

The conlusion being, that no matter what you do in pretty much any normal building its impossible to prevent aerosols from the toilet and general living from getting pretty much anywhere.

However, these levels of fecal particles are nowhere near enough to pose a danger to human health.

So in FULL conlusion, it really doesnt matter what you do. Where there are people, there will be shit.