Your sewage system is already partially private. I can say this with confidence if you live anywhere on earth outside of maybe North Korea.
It's a question of how much. Like my city for example: if you had a sewage emergency at night the people who show work for a private company. All the people working, part-time, there are pretty much retired government workers. Now is this a big deal? I personally don't think so. The same company that services the night for my city has it for the neighboring ones. It just makes sense to have this be a part time job that spans a big area vs hire someone full-time to only work in one city.
Now who built your system? It was almost certainly a private company. That subcontracted it out to other private companies. Is this a big deal? Meh, it depends. I have seen incompetence in both government engineers/project managers and I have seen it in private sector as well.
Now who maintains your system? Well again it depends. Some special snowflake equipment you are going to have to call the OEM or their rep, day to day stuff the local government employees can usually handle.
I guess my point is your system is already partially private and I am not really convinced altering it more government or less government would add any additional benefit. We have a system that basically works.