I had to present this paper for a fluid mechanics class during COVID and yes, the particles do spread. The radius of contamination was almost 1,5m.

Shared bathrooms in hospitals, rehabilitation centers, or assisted living facilities are used by patients who might be infected, thus making them a likely source of indoor cross-contamination. The pathogen-spreading potential of toilet flushes was investigated in toilets seeded with microorganisms that were later recovered from surfaces and in the air after flushing. The organisms in the bowl could not be fully cleared even after repeated flushing, and the droplets produced by flushing harbored the organisms that were used for seeding, which remained airborne and viable.

Recently, Johnson et al. (2013a) investigated different toilet designs and found that up to 145,000 sampled particles can be produced per flush.

Analysis of more recent data revealed that a large number of droplet emissions are not visible to the naked eye (d < 100 µm) (Figure 6b). These emissions account for more than 6 mL and can remain suspended in the air for a long time compared to the larger visible drops (with diameters up to 6 mm) that end up on surfaces.

The larger visible drops settle on surfaces within milliseconds, whereas the smaller, invisible drops are advected by local airflow (on the order of a few centimeters per second). Droplets settling on surfaces can be tackled in accordance with surface decontamination procedures of local infection control protocols. However, no system or protocol currently addresses air contamination. Furthermore, usual cleaning solutions not effective in neutralizing the most resistant pathogens, such as the spores of C. difficile, may even contribute to their dissemination by effectively lowering the surface tension, for example, down to 30 mN/m, compared to water at 72 mN/m, increasing the local Weber number and thus promoting fragmentation into either more or smaller droplets, depending on the fragmentation mechanism.

https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-060220-113712