O Prophet ()! When you divorce women, divorce them at their 'Iddah (prescribed periods), and count (accurately) their 'Iddah (periods ). And fear Allah your Lord (O Muslims), and turn them not out of their (husband's) homes, nor shall they (themselves) leave, except in case they are guilty of some open illegal sexual intercourse. And those are the set limits of Allah. And whosoever transgresses the set limits of Allah, then indeed he has wronged himself. You (the one who divorces his wife) know not, it may be that Allah will afterward bring some new thing to pass (i.e. to return her back to you if that was the first or second divorce).
Then when they are about to fulfil their term appointed, either take them back in a good manner or part with them in a good manner. And take for witness two just persons from among you (Muslims). And establish the witness for Allah. That will be an admonition given to him who believes in Allah and the Last Day. And whosoever fears Allah and keeps his duty to Him, He will make a way for him to get out (from every difficulty). [65.2-3]
The verses above seems to indicate that you don't need any witness to give talaq but you need witness to seal the divorce when the wife's idiah period is over. Is my assumptions correct? Is this view accepted among Sunni and Shia scholars?