We have separate bins, bags and boxes for food waste, textiles, paper, cardboard, plastics, tins, glass, and small electrical items (phew!), all collected every week. Garden waste fortnightly, but that's optional and chargeable (we have a compost heap instead). And all other waste collected fortnightly. It works well, apart from a few weeks of chaos every time the council changes supplier, when the lorries get lost or come on the wrong day, catching everyone out!
As for a rebate if you opt for the private collector, there's not a cat in hell's chance. We live on an unadopted street, and are responsible for maintaining our own road and pavement, but we still pay full council tax.