Hi again… Hopefully someone on the dev team at Monday may chip in, but I know they are working on forms at the moment. They are adding some “conditional” controls to the forms, but as mentioned, currently, forms dont support looking up data in boards (Dev team - would be good if we can tag specific boards as “public” boards and allow them to be referenced in forms!)
If you use a board, @Leon is also correct, each person needs to either be a member or invited to the board. We have some forms where we have created dropdown & status columns as lookups, but it is a pain, so where we we use the boards directly, with column permissions/board permissions to help structure/limit the details, automations to help with data validation/workflow and some formula columns to provide users with guidance on what to do next/what to fill in.
IMHO - Monday forms are great to replace basic form filling/surveys , but not (yet) powerful enough to act as true data entry, so in most cases still have to use boards as data entry…
Hope that helps