Is it possible to get HIV from sharing a public toiletWatch
Basically the only risk from sharing toilets is if you are litterally using it at the same time. The virus dies within seconds of exposure to air so even if you come into contact with any contaminated fluids (pee can't be contamiated unless mixed with another fluid btw) it would have to be pretty much direct from the source to have any impact. So even if there was anything on that tissue it would not be able to infect you because it would be dead already. Similarly you wouldn't be at risk from water fountains or anything like that.
If you are really concerned go ask a pharmacist to explain it, but multiple factors mean you are not at risk from this scenrio. Direct body to body contact and sharing needles are basically the only way it can be transmitted.