How do anarchists propose handling public works, healthcare, etc?
Well, how we do it now?
During the Spanish Civil War, anarchists socialized many industries. What they found out was that you could just remove people from most management positions and continue work as before. Very rarely a manager was actually needed, and when it was they would simply elect one of themselves to fulfill the role for a while.
Talking about healthcare specifically, in many countries that have public healthcare, the system is already decentralized. Because it needs to be, otherwise they can’t properly answer the demands from their communities. Again, you just need to remove pointless middle-men and other workplace hierarchies (like physicians being more important than nurses), and stuff tends to get better.