For the largest school within Twelver Shiʿism, the Usooli school, the fatwas of marajaʿ should be sufficient proof that witnesses are not necessary within the school. The reason I say this is because marajaʿ are dedicated to studying the Qur'an, hadith corpus, applying the principles of jursiprudence (usool al fiqh) to derive Islamic ruling. Thus, their rulings can be considered expert opinions on a particular topic.
For example, Ayatollah Sayed Ali al-Sistani (likely the most emulated scholar in the Shia world) does not require it and has made it clear in his risalah (treatise on practical law). Here are his exact words:
It is permissible for the couple to recite the formula of marriage agreement by themselves or by appointing representatives who will recite it on their behalf. There is no condition for the presence of witnesses during the solemnization of the marriage, just as the presence of a cleric is not a condition for the validity of the marriage. [Source] (Emphasis mine)
It is like that for I'd like to say all marajaʿ, but I cannot be certain of that fact.
Regardless, if you find it necessary to have available to you proof, here are some primary sources regarding this issue (note: I bold actual content of the hadith):
Hadith 1:
Ali ibn Ibrahim has narrated from his father from and Muhammad ibn ‘Isma’il has narrated from al-Fadl ibn Shadhan from ibn abu ‘Umayr from Hafs ibn al-Bakhtariy who has said the following: “About the case of a man who marries without witness, abu ‘Abd Allah [i.e. Imam Jaʿfar ibn Muhammad al-Sadiq, the Twelver 6th Imam], ‘Alayhi al-Salam, has said, ‘It is not unlawful.’” [Source]
This hadith is graded by Allamah Majelsi as "effectively sahih (lit. authentic)" which is about the highest grade you can get for a hadith. One hadith like this usually allows a jurisprudent to make a ruling if there are no countering hadiths.
Hadith 2:
Ali ibn Ibrahim has narrated from his father from and Muhammad ibn Yahya has narrated from ‘Abd Allah ibn Muhammad all from ibn abu ‘Umayr from Hisham ibn Salim who has said the following: “Abu ‘Abd Allah, ‘Alayhi al-Salam, has said, ‘Witness and testimony (for marriage) is only for the sake of children and inheritance.’ In another Hadith it is said, ‘and it is for the sake of criminal issues.’” [Source]
This was graded also graded as "effectively sahih"* and corroborates the previous hadith.
Hadith 3:
Ali ibn Ibrahim has narrated from his father from ibn abu ‘Umayr from ‘Umar ibn ‘Udhaynah from Zurarah ibn ‘A‘yan who has said the following: “Once abu ‘Abd Allah, ‘Alayhi al-Salam was asked about the case of a man who married a woman without witness. He (the Imam) said, "It is not unlawful at all between him and Allah. Witness in marriage is for the sake of children, if that is not the concern then it is not unlawful." [Source]
This hadith is graded as "Hasan" which literally translates to "Good". According to Ayatollah Baqir al-Irawani, a grade of Hasan is "one where all or some of the narrators in the chain are Twelver Shīʿa, but they have not been mentioned as just and have only been praised."** Hence, this kind of hadith is regarded as reliable, especially when it corroborates the previous hadiths.
Hadith 4:
A number of our people have narrated from Sahl ibn Ziyad from Dawud al-Nahdiy from ibn abu Najran from Muhammad ibn al- Fudayl who has said the following: “Abu al-Hassan, [Imam] Musa [ibn Jaʿfar al Kadhim, the Twelver 7th Imam], ‘Alayhi al-Salam, once said to abu Yusuf, al-Qadi, 'Allah, most Blessed, most High, has commanded in His book about divorce and He has placed strong emphasis on the presence of two witnesses. He has not accepted anyone as witness other than two just ones. He has commanded in His book about marriage but He has left it (presence of witness) as overlooked. You, however, have established firmly the need for the presence of witnesses where He has overlooked, but you have invalidated the need for the presence of witnesses where He has placed strong emphasis (which is divorce).”'[Source]
The hadith is from the 7th Twelver Imam, Imam Musa al-Kadhim, where in it, he is criticizing the practice of Sunnis who have the requirement flipped from the typical Shiʿi practice; Sunnis require witnesses for a marriage but not for the divorce. In the hadith, the Imam says that Allah specifically ordains witnesses for divorce, but not for marriage, effectively accusing the practice of not being legitimate.
This was rated as "Daʿeef" which means "weak", but given the other hadiths, effectively acts to strengthen the rule that witnesses are not needed in a marriage.
The Quran:
I wanted to explain what the 7th Imam was talking about in the last hadith, where he says "Allah, most Blessed, most High, has commanded in His book about divorce and He has placed strong emphasis on the presence of two witnesses". The Imam is referring to the second verse of the 65th chapter (al-Talaq):
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] Allāh. That is instructed to whoever should believe in Allāh and the Last Day. And whoever fears Allāh - He will make for him a way out. [65:2 - Saheeh International]
The Imam then goes on to say that the Qur'an has made no such requirement with regards to marriage. Therefore, by the principle within Islam regarding worldly matters that "all things are halal unless proven otherwise", witnesses for marriage are not necessary.
*The important part of the hadith is anyway.
**al-Irawani, Introduction to Rijal Studies, p. 46
