The contact module in Drupal 7 core uses a pair of cookies to pre-fill name and email in the form for returning users. Since the change in legislation around user details in cookies a lot of site owners will not want these cookies to be set.
Since editing core is a cardinal sin a good way to prevent the server setting these cookies is to add a submit hook for the contact form in a form alter in your own module.
/**
* Implements hook_form_FORM_ID_alter().
*/
function module_name_form_contact_site_form_alter(&$form, &$form_state, $form_id) {