This garden had been mainly ignored for years, and as well as reclaiming the parts that had been cultivated I have been creating new beds on virgin clay soil, so requirements continually outstrip supply. I compost everything I can get hold of, but I am always turning, sorting and sieving, partly to get it to work more quickly, partly to extract something I can use early on. This means it is often still full of seeds, no harm on the garden, I knock them over with the hoe as two leaves and the soil is soon good and clean, but it is a pain for planting seeds, so I do buy commercial compost as well. It varies considerably in quality, though I can't say I have found other detritus in it, however it does get heated, I don't know whether naturally or artificially, presumably to kill off things like seeds, and I have taken to handling it only with tools or gloves as it makes my hands black in a way that is difficult to get off. It's a pain, it is so much easier to make a hole and close it up around the plant with my fingers, but I don't want to scrub my hands raw all the time.
The plastic bags get saved and used until they are falling apart, I have a bunch of them full of leaves at the moment, and all sorts of stuff goes into them. Turned inside out they are black and get nice and warm in the sun, which helps decomposition, then stood with a board in front to keep the sun off they are great for growing potatoes, you can roll them down and back up aain as you earth up. At the end of their life they are filled with rubbish for the dump.
Kitchen waste goes into a compost bin stood on a piece of aviary wire, it seems rodents can't get their teeth into the curved plastic, though I have seen the occasional very small mouse that has got through the wire.
Worth saying, what goes through the compost really improves the ground underneath, worth building it where yo want to use a bit of virgin ground. Also that compost heaps are home to slugs in a big way. They help break down the compost as much as worms in many cases, but don't situate your heap within eight feet of anything you are growing, or you will find it gets 'slugged'.