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) {
   $form['#submit'][] = '_module_name_contact_personal_form_submit';
 }

/**
 * Additional form submission handler for contact_personal_form().
 *
 * Remove the cookies set by the form for autofilling user details.
 *
 * @see contact_personal_form_submit()
 */
function _module_name_contact_personal_form_submit($form, &$form_state) {
  user_cookie_delete('name');
  user_cookie_delete('mail');
}