There are different rules for different food banks. In one community centre, there is a donation twice a week from a local Waitrose, both food bank donations from customers and excess stock. These are set out on tables according to type and anyone is welcome to come in (no referrals, no registrations, everyone is welcome) and can take one item from each table.
In the neighbouring borough, the food bsnk receives funding from the local council as well as donations, is open five days a week and has many other supporting activities (health checks, welfare rights, jobseeking support). However, people have to be referred there, it isn't open door. I was interested that very few people come back for more than one parcel. I'd like to think this is because their problem was solved but I suspect it is because it is too much to go through.
Other places, especially churches, run kitchens where people can get a meal. Although the funded projects probably meet the immediate or emergency needs of more people, the community groups provide more long-term support and the opportunity to form relationships that support people more widely.
When I volunteered with one that was connected with a health org, they often got donations of day old pastries from Starbucks and they literally had freezers full. So after getting their more healthy food, people could take as many pastries as they wanted because there were always mountains. At a food bank I was a client of a couple of times, they got day old pastries from a grocery, and again, mountains of slightly stale pastries, only less healthful food overall because they didn't aim for the health needs like the health org one did. It was pretty much the diabetes food pantry there.