(Rodney Hide) listed these (core functions) as transport and water services and public health and safety, such as rubbish collection.As it stands, I don't think the current setup is ideal.
Why do they provide rubbish collection at all? Presumably there is a public good argument - an untidy town is detrimental to all, but there is an incentive to free-ride on the cleanliness of one's neighbours. However, at least in Wellington they make you voluntarily pay for rubbish collection through buying rubbish bags. So it seems like they put the incentives right back there, which kind of defies the point of centralised bodies providing public goods.
However I suspect there is a greater complication - they also clean the streets, parks and pavements. If private firms collected your rubbish, you could just get the same thing for free by dumping it in the park and getting the council to pick it up. So the private firms would go out of business, and the council would have a whole lot more work to do cleaning the parks. If you don't think people would do that, you haven't seen the streets around the student flats in Otago University.
However, we should note that these incentives still exist as long as the council makes us pay for rubbish bags.
If rubbish-collection is a public good, it should probably be free. Otherwise it should be privatised. I can't really see the benefits of having a user-pays, centrally provided system like this.