This was a strange one on first inspection. We have a group of users in another location that used a different AD group from everyone else for accessing our various SharePoint sites.
For all the sites we'd opened up to them, they could see items and open the relevant nintex form. However on our news site, they could see all the news items but upon opening the form would get a request permissions to this page.
My first thought was to some deeper permissions but as they could open other nintex forms, I looked at the connected lookup lists. I've had this before where if you haven't given read rights to the lookup list and you're displaying a field from that list it on your form, it won't open. That wasn't it.
It turned out to be the audience ( people picker ) field. I'd had an issue were somehow I've ended up with All Users as an option twice. This would make the form unable to save when choosing all users. To get around it I creates a SharePoint Group with all my users and required groups within and limited the field to that group.
Looking at the group I found the issue.
Because I had set the 'view membership of the group' to be Group members only, the group having the issue,who weren't part of this group, were unable to see the audience field when opening the form. Setting it to everyone solved the issue.