TWH: It seems pretty obvious that Paul used an amanuensis at least once. At the end of his letter to the Romans, we find the following sentence: "I, Tertius, the one who is writing this letter, greet you in the Lord" (Rom. 16:22). You can't really get around that one. It's interesting. It makes sense that someone like Paul would use an amanuensis at times. For example, the conditions during his imprisonment seem to warrant the use of one. It's more difficult to ascertain whether Paul used amanuenses elsewhere in the New Testament.
Let me give you a couple of quick tidbits of info:
1. There was actually an Old Testament amanuensis. Do you know what his name was and for whom he wrote? You'll find him mentioned in Jer. 36:4–6 and 45:1. His name was Baruch, and he wrote for Jeremiah. It's interesting that Jeremiah is told to make use of an amanuensis. As far as I know, he's the only prophet who is.
2. There are a few Greek words that are sometimes translated as "amanuensis": προχειροφόρος (also προχειράριος), ὑπογρᾰφεύς, χειρογράφος. There may in fact be more, but these are the ones that come to mind right off the top of my head. You'll recognize that last word, I bet, from Col. 2:14, where Paul uses the neuter form χειρογράφον. That word in Colossians refers to a "certificate of debt" (for my distaste for any translation involving the word "handwritten," please see here).There are questions surrounding whether Luke served as an amanuensis for Paul, or Sosthenes, or Timothy. Some speculate that Mark served in this capacity for Peter, others Silvanus, etc. Did these men make use of amanuenses, specifically Paul, and, if so, what sort of role did an amanuensis have on the actual words of the New Testament? That, my friends, is a seriously debated question.
The reason that scholars debate the issue of amanuenses so much is because of the differences encountered in texts by the same author. For example, why does Philippians or Romans looks so different from 1 and 2 Timothy when it comes to vocabulary, syntax, and rhetoric? In fact, it's based on these arguments that many––over the past couple hundred years––have written those letters off as not written by Paul. Discussions like this have taken place since nearly the very beginning of the founding of Jesus' church. If you do a study on when New Testament texts were accepted as part of the canon, you'll see a debate of sorts pertaining to 2 Peter. It's like for a period of time, certain groups loved it because it was super interesting, but did not believe it was written by Peter. All of this because of the difference in style––vocabulary, syntax, and rhetoric––between 1 Peter and 2 Peter. By the way, in modern times, that's the big reason people have shunned Hebrews from the Pauline corpus, which is where it belongs and where it was for most of the history of the Christian church. What scholars have failed to recognize is that authors have a wide stylistic range, much like certain syntactical constructs (e.g., the simple genitive) have wide range and certain lexemes have wide semantic range. The authors of the New Testament should be allowed to have different style depending on many factors (e.g., what side of the bed they woke up on, to whom they are writing, from where they are writing, how urgent their discourse is, etc.). Nevertheless, the differences in style have taken scholars to places for many there is no path back and most probably would rather never come back. It makes more sense, to them, to just say they are not written by the author whose name is affixed or whose name history has consistently ascribed to said text.
Some have suggested that Luke served as Paul's amanuensis based on 2 Tim. 4:11a, which reads, "Only Luke is with me." If that's the case, then Luke must have served as the amanuensis––so the argument goes. Though, with that said, people have proposed Tychicus instead of Mark (e.g., Jeremias); that of course hinges on the verb ἀπέστειλα being an epistolary aorist––and that Paul wanted and/or needed an amanuensis. Now, I'll grant that Paul was alone during this time in his life and ministry. We know what is said in 2 Tim. 1:15: "You are aware of the fact that all who are in Asia turned away from me, among whom are Phygelus and Hermogenes." He felt completed abandoned. But I'm not sure we can make a claim that based on 2 Tim. 4:11a, Luke served as Paul's amanuensis. He could have, but I'm not sure he did or that it was absolutely necessary. Consider for a moment was C.F.D. Mouse said about Luke and his relationship to 1–2 Timothy and Titus:
"Luke wrote all three Pastoral epistles. But he wrote them during Paul's lifetime, at Paul's behest and, in part (but only in part), at Paul's dictation" ("The Problem of the Pastoral Epistles: A Reappraisal," 434).Therein lies the issue. What in the world is going on? Did Paul write 1 and 2 Timothy and Titus or did Luke? Did Paul tell him what he wanted to say, give Luke some talking points, and then allow Luke to fill in the parchment? Did Luke write the a letter first, give it to Paul, let Paul mark it up and give his feedback, and then have Luke rewrite it? What's going on? That's what scholars debate regarding the style and authorship. And the use of different amanuenses makes it easy to explain the differences between texts by the same author.
I mentioned earlier that the proposal that Tychicus acted as Paul's amanuensis hinged on the send-verb being an epistolary aorist and that Paul "wanted and/or needed an amanuensis." I can see why Paul would use an amanuensis for his letter to the Romans. It was longer, much longer, than his average texts. But that doesn't necessarily mean he needed or wanted an amanuensis every single time he wrote a letter. 1 Corinthians ends with Paul mentioning to his audience that he is writing a portion in his own hand. That portion is the "greeting" (ἀσπασμός, 16:21). This refers to the portion of the text that begins there in v. 19 (where the word "greet" appears four times) and extends to the end of the letter. Why mention his own hand unless the original letter featured two different scripts, one belonging to the amanuensis and the other––at the end of the letter––to Paul? But the use of the amanuensis here could be attributed to the length of the letter. When it comes to a letter like Philippians, there is nothing within the text to suggest that Paul utilized an amanuensis. Some, to be sure, have proposed Epaphroditus for this position, but where is the evidence beyond the fact that Paul sent the letter by way of Epaphroditus? I can't find any evidence in the text beyond that, and I would caution people to go only so far as the text will permit.
I might also point out another reason why people have argued that Sosthenes in particular functioned as the Corinthian amanuensis. In 1 Cor. 1:18–31 and 2:6–16, you'll find the use of the first person plural ("we" and "us"). Take a look at those texts when you get a chance. Paul identifies himself in the letter opening as the author. He also includes the name "Sosthenes. It's important to note, however, that in the one particular text where we know for an absolute fact that an amanuensis actually wrote the letter (τὴν ἐπιστολήν) Paul did not mention another person's name in the letter opening. In Romans 1, you will find Paul's name and only Paul's name. A better explanation for why Paul mentions a person's name in the letter opening is because of the content of the letter and because of the personal relationship that the co-laborer or brother had with the letter's recipients.
I've sort of run out of time when it comes to the last part of the question at hand. Did an amanuensis write down the very words of the author? Did he transmit only the ideas of the author? Did he edit and improve the grammar and style before sending the document out? Did he translate the author's work (e.g., from Aramaic to Greek). Etc. There's a lot more to think about here, but I'll go ahead and show my cards. I opt for the former when an amanuensis was actually used. When it came to Paul, he told the person holding the reed what to write and exactly what to write––except, perhaps, when Tertius sneaked a little comment in at the end of Romans (though I could see him asking Paul if he could, and Paul nodding his head and saying, "That'll be fine"). I don't see a lot of editing––if any at all––going on when it comes to these New Testament letters. These letters were usually written out of a serious need. There was a problem in a given church (or, in the case of Galatians, a group of churches in the same region), and those letters needed to be written and carried off as quickly as possible to address said issues.