I can answer the Vajrayana/Tibetan side of this in much more detail than the Chinese side of your question, so hopefully a Sinologist will come in and fill in the blanks regarding how Yijing and Xuanzang's process worked.

The long and the short of it is that we don't really know the method with which Buddhist translators translated scriptures from Indian languages. We know things like who translated them, where they were translated, and which Indians assisted the Tibetan translators, but the philological process in the 9th and 11th centuries is more or less lost to us.

It's important to get a bit of a timeline straight, because calling scriptures translated "at an early date" skews the perspective a bit here.

The earliest Buddhist scriptures, the Sutras and the Vinaya, were set down to writing in the third century B.C.E. At the time, they were written in both Pali and Sanskrit (as well as other languages, allegedly). The Pali scriptures survived and form the basis of Theravadin schools in Southeast Asia. The Sanskrit scriptures were the ones that would get translated and used in Mahayana school (i.e. the ones in China, Tibet, and their successors). I'm not sure (and I'm not sure how sure other scholars are) exactly how much difference there is between the scriptures or even how much the languages differ (I don't know a lick of Pali, frankly) but I know that some scriptures are exclusive to one or the other. For example the famous Milindapanha (The Questions of King Milinda) is extant and held high in the Pali Canon, but is apparently significantly shorter in the Chinese Canon, and is completely absent from the Tibetan.

The later texts, the Abhidharma, the Shastras, and the Tantras came to being over the next millennium. So naturally, there's a lot of linguistic variation between these three genres of literature and their authors (despite often times having the same name, i.e. there are clearly two Nagarjunas, though distinguishing one from the other is difficult. The first is usually considered the more famous one who composed the Mulamadhyamakarika, while the second is usually referred to as the "Tantric Nagarjuna"). Anyway, the first Nagarjuna composed his most famous work, the foundation of the Madhyamaka school of Mahayan philosophy, in the first or second century C.E. At the other end of the spectrum, Kawa Paltsek (Tib: ka ba dpal brtsegs/bka' ba dpal brtsegs) was one of Tibetan Buddhism's early translators, a student of Padmasambhava and Shantarakshita (i.e. mid 8th Century). Nagarjuna composed in Sanskrit, while Kawa Paltsek's native tongue was Tibetan and his original compositions were in Tibetan, though he was also a lo chen (literally a "great translator") who was well known for some of the earlier translations of the Buddhist Canon from Sanskrit into Tibetan.

Paraphrasing Ronald Davidson's Tibetan Renaissance, these early translators, i.e. Kawa Paltsek and his contemporaries were "better" at their job than their successors. Part of this is probably due to the Emperor (and their patron) Tri Song Detseun's insistence that the Tibetan language be standardized according to Indian terminology. I.e. that Indian Buddhist vocabulary have appropriate Tibetan equivalents. I mention this only because the later Tibetan translators of the "Renaissance" period (Davidson's classification, 950 C.E. to 1200) tended to translate less, falling back on Sanskrit terms mid sentence, leading to many bizarre Tibetan language constructions in retrospect, especially considering Tibetans' poor approximation of the much more complicated Devanagari script's nuance in terms of phonetic construction.

A small example:

The Indian scripture at the front of the Narthang Kanjur (the Tibetan Tripitaka) is the Vinayavastu. Tibetan has no "va" sound, so the Tibetan approximation is the following:

rgya gar skad du bi na ya bastu

"rgya gar" = India "skad" = speech "du" preposition

So literally, "In the Indian speech, 'Vinayavastu'"

And then is followed with,

bod skad du 'dul ba gzhi bam po dang po

"bod" = Tibet "'dul ba gzhi" = Basis of Moral Vows, i.e. Vinayavastu "bam po" = section, volume, or chapter "dang po" = first

"In the Tibetan speech, the Basis of Moral Vows, Ch. 1"

This is purely hypothetical and could probably form the basis of a Ph.D. thesis, but it makes me curious as to the form or formlessness of Sanskrit during the translation periods, since the Tibetans refer to it universally as "the Indian language" which seems rather... general for such experts of language. It shouldn't be so bizarre of a concept considering that Tibetans were going at least to Nepal, Magadha, Bengal, and Kashmir, and any brief survey of Indian languages reveals a cacaphony of tongues in that geographic breadth, so it seems to me that Tibetans of such learning capacity were unable to recognize that they were different. Anyway.

In some colophons of these texts (mostly of the later diffusion variety) there are even more interesting and useful descriptions. Turning back to Renaissance, Davidson quotes the colophon of several texts (identical),

The Magadhan Pandita Anandabhadra and the Tibetan translator Setsa Sonam Gyeltsen translated, edited, and finalized this in accordance with a Magadhan text in front of the self-originated Visuddha-stupa to the south of the great city of Tirahati. Later, Tarpa-lotsawa, the Sthavira Nyima Gyeltsen, correctly translated it according to the instruction he heard from the siddha Karnasri in Sri Nalanda-mahavihara.

We're quickly veering away from areas I can talk about with more specificity, and I'm willing to guess that most historians and philologists at this point have mostly educated guesses. While these translators, both the early Royal ones and the later Renaissance ones, spent a great deal of energy and effort and did awesome work, they were pretty lax with recording their process. Some scholars of Gedun Choepel might know a little bit more about his process since (I think) he kept journals of his time in India, and he's well known for his compositions (The White Annals, Grains of Gold, and The Tibetan Art of Love to name a few) and translations (The Dhammapada being the most famous) however Gedun Choepel worked in the 20th Century, hence our comparative knowledge of him, his method, and also the unreliability for being able to compare him with his distant predecessors.

It should be noted at this point that while Tibetan translators were much more heavily reliant on India and Indian scripture for their base of Buddhism, their weren't totally independent of China and Chinese sources of inspiration. The Tibetans weren't introduced to Buddhism until the 7th Century and didn't start making their own translations until the 8th. Buddhism's entry into Chinese intellectual life started much earlier, was much slower, and started at the fringes of society (i.e. groups that made their living on the Silk Route). And it was through this economic-based diffusion that eventually led to the first translations, and later monks like Yijing and Xuanzang to travel to India, compose their famous travelogues, and return via the sea route to China with their masses translations. This is also one of the reasons why the Chinese Canon has many doubles (a similar set of events is why there are also many doubles in the Tibetan Canon).

In the Tibetan Canon there are two versions of the Lankavatara Sutra. One that scholars hypothesize comes from the Sanskrit original, and one that scholars think comes from the Chinese translation. Only a thorough linguistic analysis leads to these kinds of conclusions, and they're the kinds of analyses that Tibetologists and Buddhist Scholars are working on all of the time because these Canons are just... just massive.

con't.