I think the answer is"it depends".
Some allotments are private. Some are council run. Ours is a slightly unusual type. It is a council site, but day to day running is done by an elected committee.
So the committee members have to organise rent collection, which is then forwarded to the council. They also have to ensure tenants adhere to various rules, and as and when necessary arrange evictions. There is a shop on site so that requires stock, and a rota of volunteers to be organised so it can open each Sunday. Finally the committee have to hold meetings including an annual general meeting that all plot holders can attend, and get to vote on who is on the committee and other decisions about the running of the site.
Our committee has ten members in all, but most of the work falls to just four.
Field manager: Lettings, rent collection, gate keys and security.
Treasurer: All financial transactions, book keeping, everything financial, getting everything audited at the end of the year.
Shop manager: Buying stock, organising the volunteer rota, taking and supplying seed orders.
Secretary: All paperwork, all correspondence, both external and with plot holders, booking meeting rooms, taking meeting minutes. Liaison with council.
My guess would be that most sites have to undertake a similar range of tasks and responsibilities. But I would also guess that defining exactly what is done by the plot secretary will vary enormously, and could be very different from one site to another.