First, the user will need to be assigned to the the Smart Fields system role.

Once this is complete, the Smart Field instance must be assigned on the Smart Fields tab of the Smart Fields role:  
  1. Go to Administration > Security > System Roles > Smart Fields
  2. Click on the Smart Fields tab
  3. Click "Assign smart field instances"
  4. Browse to the specific Smart Field type and highlight it
  5. Check the box on the right for the appropriate Smart Field and click Save, the users in that role should now have rights to see the fields