(I originally asked the same question but it was answered only by Shi'a brothers. Since there were no Sunni answers, I accepted that answer and opened this new question requesting a Sunni view.)
Why Sunni Islam does not require witnesses for divorce? It is such an important turning point in one's life and it must require witnesses. There are cases where couple live together even though the husband has issued verbal divorce. But they still live together because of either shame or the difficulty they have to go through if separated. Shi'a Muslims refer to this verse to justify that witnesses are required in divorce:
Thus when they fulfil their term appointed, either take them back on equitable terms or part with them on equitable terms; and take for witness two persons from among you, endued with justice, and establish the evidence (as) before Allah. Such is the admonition given to him who believes in Allah and the Last Day. And for those who fear Allah, He (ever) prepares a way out,