In Sunni Islam, I am often told that divorce can be performed without witnesses, which doesn't really make sense to me since everything in Islam requires witness (even when an adulterer commit adultery require witnesses). Is this the same in Shi'a Islam, and if so, why? Wouldn't unwitnessed divorce cause problems?
According to Qur'an divorce requires witness: At-talq:65/2.
A man can start the process of divorce in presence of no one but it should be completed and committed with two witnesses after they fulfilled their term otherwise the divorce is not done and they return to their normal life.
Shi`a and Sunni Muslims differ in the acceptable method of divorce. Wikipedia says Sunni practice requires no witnesses and allows triple-talaq and Shi`a rules do not allow this.
The primary evidence used by the Shi`a are the first two verses of Surat at-Talaq:
O Prophet, when you [Muslims] divorce women, divorce them for [the commencement of] their waiting period and keep count of the waiting period, and fear Allah , your Lord. Do not turn them out of their [husbands'] houses, nor should they [themselves] leave [during that period] unless they are committing a clear immorality. And those are the limits [set by] Allah . And whoever transgresses the limits of Allah has certainly wronged himself. You know not; perhaps Allah will bring about after that a [different] matter. [At-Talaq: 1]
And when they have [nearly] fulfilled their term, either retain them according to acceptable terms or part with them according to acceptable terms. And bring to witness two just men from among you and establish the testimony for [the acceptance of] Allah . That is instructed to whoever should believe in Allah and the Last day. And whoever fears Allah - He will make for him a way out [At-Talaq: 2]