I just remembered, a farm i visited a few years ago had theirs on concrete, actually in a set of concrete "pits" (think 3 waist-high walls and a floor in each), but it was dug into the ground, a bit like a silage tunnel if you're familiar with that.
The farmer was doing large scale composting, lots of manure from various animals, occasional carcasses, got wicked hot. He was using a
tractor to regularly turn the piles, and the concrete facilitated that (the site was hilly, clay, and had heavy rain, also strong large animal pressure, including large monkeys, goats, things that make a right mess out of a big pile). He also was able to leave the piles for a good long time (a year or more), and the wildlife eventually found their way in.
I think a big factor is whether/how you may have to turn your compost.