I do some of both and really do not know which is best. I store some, about as much as I think I will need for seed starting and various projects, by putting it on concrete and covering it with something to keep the rain off it, or in some big garbage cans with lids. As long as it is moist, it will keep working and breaking down, let dry out it seems to go dormant till it is wet again.

The majority, I just keep on adding greens and browns to and continuing to build the pile. I think that it being in there helps to keep breaking the new materials down, but that is just my homespun opinion on it. I try to time it so that the majority of fall leaves have been composted, or shredded and set aside for mulch in the spring, and layer the finished compost on the garden in mid winter when I turn under the green cover crop and add a layer of horse manure.

This seems to work well for me in my location and set of circumstances. I am sure that many other variations work as well.
Bill P.

If you have finished compost now put that on your garden and start a new compost pile with any material you can for next year. Storing compost in the garden is the best plce for it since the soil bacteria will start to incorporate that compost into your garden so what it has will be available to the plants next growing season. Waiting until spring to put compost on the garden means that what is in that compost will not be available to plants until the year after.

Yes, I agree. For one thing, the garden is a really handy place to store compost or dirt or anything that is eventually going to end up in, well, the garden. For another, it frees up the space to start a new pile, which is always kind of fun.