Most of the public facilities in my town are equipped with bidet/jet + toilet paper (some might provide soap at the toilet, if not; they always provide soap in the hand sink. In this case, it's better not to touch your private parts with bare hands and use the toilet paper to fully clean it). All of the office that I went to always provides soap and toilet paper. So you don't have to bring your own kit.
The shape of the toilet and the positioning of the bidet/jet definitely could reduce the amount of water going outside the (pardon) hole. But there's always some of it that got out. It is advised that we wipe the water on the seat with toilet paper to make it dry. Some also provide a cleaning solution to wipe the seat before using it.
Usually, there's a cleaning service nearby that always wipe the floor and clean everything.
From my personal experience with toilet paper, I need to use a load of it to make sure its thick enough to protect my hands from touching it directly as there's no water to wash it later except outside in the hand sink. I could easily use half of the roll to fully clean myself and make sure there is no residue at all (further wipes doesn't leave any mark behind the paper). Also, we don't throw the paper into the (pardon) hole, there's a dedicated bin for it and the smells (pardon) when it is opened is a whole new problem (I don't know if the gas is dangerous, but its really something). But then again, as it is not cleaned with soap, I am sure there are some bacteria left behind and it makes me feel uncomfortable.
I don't know what to say about the perception of people, but I do think it does change the perception, I am just not sure how :)