Problem: TinyMCE rich-text editor disables the operating system spellchecker when using Firefox or Chrome.

Solution: Set the TinyMCE configuration “gecko_spellcheck” to TRUE.

tinyMCE.init({
	...
	gecko_spellcheck : true
});

In Drupal, you can override this setting by adding an alter function to a custom module.

function MYMODULE_wysiwyg_editor_settings_alter(&$settings, $context) {
  if ($context['profile']->editor == 'tinymce') {
    $settings['gecko_spellcheck'] = TRUE;
  }
}

Source: http://drupal.org/node/1064268