Approximately 67% of the hadith in the Imāmī corpus go back to Jaʿfar b. Muḥammad al-Ṣādiq (d. 148). The advantage of this is that Jaʿfar was teaching hadith at a time when writing down reports was a default not the exception. This means that there is no lengthy oral period of transmission to contend with.
There is a lot of incidental evidence that can be used to demonstrate this but I will present just one: The case of Jaʿfar’s student called ʿAbdallāh b. Sinān.
We know that ʿAbdallāh b. Sinān was writing down what he heard from Jaʿfar because he reports the following:
I asked Abā ʿAbdillāh about a woman who breast-feeds a slave boy of hers, born to a female slave belonging to her, until she weans him. Is it permissible for her to sell him?
He replied: No. His price is forbidden to her (i.e. she cannot sell him). Did not the Messenger of Allah say, “It is forbidden through suckling what is forbidden through descent”? Has he not become her son?!
I started writing it down whereupon Abū ʿAbdillāh said: “The likes of this are not written down”
وعنه عن الحسن بن محبوب عن عبد الله بن سنان قال: سألت ابا عبد الله عليه السلام عن امرأة ترضع غلاما لها من مملوكة حتى تفطمه يحل لها بيعه؟ قال: لا، حرام عليها ثمنه أليس قد قال رسول الله صلى الله عليه وآله: يحرم من الرضاع ما يحرم من النسب!؟ أليس قد صار ابنها! فذهبت اكتبه فقال أبو عبد الله عليه السلام : وليس مثل هذا يكتب
Note that Jaʿfar was not against the writing down of reports, but this question was such a basic one (everyone should know that relation by fosterage is akin to that by blood) that he did not deem it important enough to warrant writing.
We also know that ʿAbdallāh b. Sinān had begun transmitting his book containing the reports of Jaʿfar to his students. He once asked Jaʿfar:
The group comes and hears your hadith from me, but I (sometimes) become weary and do not have the strength (to read out all the hadith for them) (what should I do)?
He said: Read out for them from the beginning (of the book) a hadith, from the middle a hadith, and from the ending a hadith.
وعنه، عن أحمد بن محمد ومحمد بن الحسين، عن ابن محبوب، عن عبد الله بن سنان قال: قلت لابي عبد الله عليه السلام: يجيئني القوم فيستمعون مني حديثكم فأضجر ولا أقوى قال: فاقرأ عليهم من أوله حديثا ومن وسطه حديثا ومن آخره حديثا
This speaks to formal transmission practises wherein students come to a teacher, copy down the contents of his book for themselves, and then hear the contents being read out loud by the teacher (qirāʾāh) to check for any errors.
How were ʿAbdallāh b. Sinān’s books received? Let us refer to Najāshī’s (d. 450) entry to find out:
ʿAbdallāh b. Sinān b. Ṭarīf. A mawlā of the Banī Hāshim. It is said the mawlā of the descendants of Abī Ṭālib and it is also said a mawlā of the descendants of al-ʿAbbās. He was a treasurer for al-Manṣūr, al-Mahdī, al-Hādī, and al-Rashīd. Kufan. thiqa. One of our companions (i.e. an Imāmī). Meritorious. Not criticized for anything. He narrated from Abī ʿAbdillāh ... He has Kitāb al-Ṣalāt which is known as ʿAmal Yawm wa Layla, a bigger Kitāb al-Ṣalāt, a Kitāb collecting (reports on) all the different topics on the ḥalāl and the ḥarām. These books were transmitted by a very large number of our companions because of his lofty status in the sect, his trustworthiness, and his merit.
عبد الله بن سنان بن طريف مولى بني هاشم، يقال مولى بني أبي طالب، ويقال مولى بني العباس. كان خازنا للمنصور والمهدي والهادي والرشيد، كوفي، ثقة، من أصحابنا، جليل لا يطعن عليه في شيء، روى عن أبي عبد الله (ع)، وقيل: روى عن أبي الحسن موسى (ع)، وليس بثبت. له كتاب الصلاة الذي يعرف بعمل يوم وليلة، وكتاب الصلاة الكبير، وكتاب في سائر الأبواب من الحلال والحرام، روى هذه الكتب عنه جماعات من أصحابنا لعظمه في الطائفة، وثقته وجلالته
What happened to ʿAbdallāh's books? While the books do not survive independently, reports taken from them were incorporated into later works. Thus we find that close to 1000 examples of ʿAbdallāh’s reports on legal topics are quoted in the 4 main books of Shia hadith.