I wouldn't describe the primary quality of Dat as "anonymity." It's anonymous in the same sense that, if you setup an HTTPS server without a domain name and then never share the IP, people are unlikely to find it. Dat is anonymous in that sense, though also it's harder to find, because the addresses are 32-bytes in size (way harder to guess).
The main value of Dat is how it makes publishing *very* easy because you don't need servers. One click to make a website.
But, to your main question:
Dat doesn't automatically push any data to other users. You have to navigate to a site to download it, and then it'll be temporarily cached (and removed eventually) if you don't explicitly save it. People who are doing nasty things should not get into your life via Dat.
prf