Dunno about your part of the world, but up here, a number are run by the local parish councils. In that sense, they are run like small independant sites and are not listed on the "big" council lists. they are individually administered and allocated.
For ours, we had to apply for a grant each year for funding from the parish council as a community amenity project. This was used to pay the landowner ( British Rail leased the land to the parish) a pepper corn rent and to pay the water bill for the site. everything else ( not much, only 20 plots) we organised ourselves.
Not sure that's what you meant. The other options I can think of are those run by some kind of charitable scheme or commercial letting, which would be costly. Even our small space cost plenty to run each year and we didn't have to pick up any kind of public liability insurance etc.
We were almost closed down once because a member of the public who drove past our allotments made a formal complaint that the hedges were too high & causing a nuisance. The parish council had to 'sort it' within a small number of days & as they had lost track of who the reps were, paid a commercial company £400 to do it. The ructions that ensued took ages to sort out.
In Cheshire, the high cost of land makes it difficult to get new sites started.
