Thanks for the update and sorry about the delay in getting back on this.
During troubleshooting, I noticed that although guests (non-logged-in users) do have access to both these pages, the forms on the pages are showing "not authorized message".
Please go to WP Admin -> Toolset -> Access Control -> Toolset Forms and you'll see all the post forms under the "Post Forms Frontend Access" heading.
To allow the guests (non-logged-in users) to use these forms "Vakexpert toevoegen" and "Project toevoegen", make sure that the checkbox for the "Guest" role is checked, in their column.
( example screenshot: hidden link )