You will find those who have composted for a while usualy prefer constructing their own. Personaly,I find most store bought composter far to small,not to mention needless expense. The envioramental empact of manufacturing is something I prefer to avoid by repurposing material for bins. That is certainly not to say you will find no help. Please tell us how much space you have,what materials you plan on composting,how much finished material you need and why you decided to compost in the first place. You will get plenty suggestion for diy as well as a few proven comercialy available units.
I have a mantis drum composter and with the right material I can get about 5 cubic feet of compost every three weeks. With no odor or bug problems but, it's hard to feel up the drum all at once. So if space is limited try a drum composter.
More information on our composting need: We are a community garden that has 70 raised beds 12' x 4'. 95% of the crops are vegetables of all kinds, of course including tomatoes in most beds. For years, we had six compost bins constructed atop the soil -- 3 at each end of the garden. Each bin was approx. 4'x4'x4'. In adjoining sets of 3 bins. The bins were built of chicken wire or plastic webbed fencing. We did passive composting in these bins. But in the last couple of years the rat population multiplied in our urban garden: the rats were nesting underneath the compost bins. We have removed the compost bins and now looking for a means to compost the garden waste. Not so much for the compost as to avoid sending large amount of garden waste to the landfill. We have ample space reserved for compost bins or composters, but probably not the people power to build our own rat-proof composters, unless there were a simple, easy design. Some of our gardeners have recommended that we purchase rat-proof composters. But others wonder whether purchased composters will be large enough handle the considerable amount of garden waste from the raised beds. Some wonder whether we will be able to educate gardeners on what and what not to put in a composter. Another concern is how much work will it be to "manage the composter" and who will do that? Thanks for any suggestions.
I would recommend using some kind of wire mesh or fencing underneath and surrounding the compost. I've used chicken wire, although the wire is thin enough that it will rust away in a couple of seasons. Something heavier like galvanized fencing (maybe recycled chain link if you can find it and the holes are not too big). They also make a plastic-coated wire mesh specifically for placing under compsot bins.
You can also make the bins out of stacked concrete blocks, much harder to get in through the walls. Or use wood slats but again line them with wire mesh.
Or try an electric fence wire around the outside of the compost area, at rat height? They have solar powered ones. I don't know that this will work, just a thought.
Hire some scouts with slingshots for nighttime rat patrol? Maybe they can get a varmint control merit badge.
I think you can continue with the compost piles the way you have them. Nature's animals will seek out ways to live and protect themselves, including coming close to humans.
Also, I would think a community garden would be encouraging composting. I know some gardens do make a requirement of time donated to help maintain the gardens. This could be a way to help the people be involved in maintaining this compost pile. If they are the ones who have to come in and turn it or remove bad things from it, they will know which bad things not to throw into it.
Rats or other rodents will be near the gardens and you may have to control them as how the recommendations , usually with traps.
At the same time, you mentioned that no one is actively handling the compost piles. Maybe it's time that someone does, since on a smaller scale, one of the ways to keep rodents out of a person's backyard compost pile is to keep turning it and disturbing it (of course, a homeowner would also find ways to get rid of mice and rats and not let them be a part of the "natural" list of animals running around the yard)