There was no such thing as a flushing toilet in Ancient Greece. Remarkably, there was a flushing toilet 1,200 years before that, at the Palace of Knossos, in Minoan times, and it's the oldest known flushing toilet in Europe. It probably worked by having a slave pour buckets of water into the drain.
But in Classical times, when Nico and Diotima are at work, they had no plumbing into the home. All water was carried in from public fountains, and that was intended for drinking and washing. If you needed to go to the toilet, well, that was what the chamber pot was for.
If you lived in the city, then the bad news was that there was no garbage collection service. There was however a drain that ran down the middle of every street. That's where the contents of the chamber pot went. I've made use of this fun fact without any mercy for Nico. Whenever he gets knocked down in a street fight, he invariably goes straight into that drain.
In fact we can be quite certain that's where the waste went, because eventually the Athenians passed a law forbidding citizens to dump their waste in the street. The same law created the world's first public landfill site outside the city walls (another first for Athens!) and required all rubbish to be dumped no closer than that. However this all happened in 400BC, sixty years too late to save Nico from going into the poo.
Now as to the delicate problem of a world without toilet paper...you won't be surprised to hear that this is not a well-documented subject. The Romans famously used a sponge tied to the end of a stick. The Greeks might have used a sponge too, when one came to hand. But there's evidence to suggest that a handful of clay was more common. An interesting alternative was the leaves of vegetables such as leeks.
I must mention in passing that in the absence of washing powder, the next best thing to keep your clothes clean is urine. (It's acidic.) They actually had collection jars to store it in.