Do we need seperate or indeed any privacy notices for forms filled out by staff. These forms are not processed by any 3rd party. they are internal only and are for the purpose for someone to fulfil their role. Like a form requesting permission to access a folder.
You need to consider the purpose of data processing, not the mean. Why do employees fill these forms? If they fill the form to perform their job task, then, the processing is included in the employee privacy notice which is attached to the employment agreement (if not, you should).
If the form is a survey to better understand the education or expertise of your staff it may fall under the legitimate interest of the data controller (art. 6 lett. F) GDPR)
In case the processing is not included in the employee privacy notice you can decide whether to make a specific privacy notice or to implement the employee privacy notice.
Here you can find more information on the legal basis to process personal data according to the GDPR: