What's the intent of the question? Is it a pollution/greenhouse gas (GHG) concern? Or a scientific question?
Good composting needs lots of oxygen. It will decompose better uncovered (faster, more completely, better for the plants it will feed), as long as it's warm enough. In regions with cold winters, the process will stop if it gets too cold, and covering it for the season is usually necessary.
In terms of GHGs, the gases release are minor compared to what they absorbed in their lifetime, and using it as fertilizer is much better than alternative fertilizer (which must be produced and transported).
As a scientific question I can see the interest, as a practical, sustainability question, e.g. re: GHGs, please compost as much as possible. Turning the compost helps the oxygen absorption, but also distributes the bacteria throughout to maximise their efficiency (so they always have fresh food). Most of the bacteria is below the surface layer where they are protected from the elements, and have sufficient moisture and correct temperatures (not too hot, not too cold).
Composting also releases heat - the bacterial breakdown of the cellulouse is "exothermic" - but it's not a concern for global warming. I don't mean this to be sarcastic, but rather that it terms of sustainability (and my confusion regarding the question) composting is very good either by the fertilizers it displaces, or the pollution from the systems to dispose of the material otherwise. Decomposition will happen if you do it or you let nature run its course). Best composting needs lots of oxygen (and nitrogen).