I'm curious as a tourist what happens if you don't have a local contact? What happens if one has nothing but a hotel booking for accommodation instead of a home? I'm guessing that wasn't your case but were there others you heard or saw who were just regular tourists?
Actually my local contact was simply my Airbnb host. I had arranged the Airbnb only earlier that day, In response to the new travel restriction.
Other passengers provided the number of their hotel. The CDC called the hotel and (I believe) checked whether they were aware that these passengers were about to self-quarantine there.
Edit: even if you arrived without having anything booked at all, you can pay to stay at a designated quarantine hotel, though that is more expensive.
Also note that Taiwan doesn’t grant the “visa exemption” usually granted to tourists, when they now arrive from countries assessed as level 3 (requiring home quarantine). I was able to enter during a short window of time where home quarantine was instated but the new visa rules were not yet in effect. So I think it isn’t usual for the situation you described to occur.
I'm curious were you traveling around Asia anyway and decided to visit Taiwan due to the Coronavirus being very contained there? Are you seeing lots of other travelers? Is your plan to wait it out there if so? It doesn't seem like a bad place to be right now.