You need to open the form in design mode.
If users use a button within the form to submit, double click it and view the properties.
If it is set to rules and code then you would need to check the rules and or code, whichever it contained.
If the button says Submit or if users user the submit button in the toolbar then Go to Tools-> Submit Options.
Look at the items which are included and like I said add meesage box actions if needed.
Definetely it is a permissions issue but you need to diagnose exactly where. My gut tells me that if the form is being saved in the library that the error is coming from something else.