My story may be different than most. I took Greek Prose Composition during my second B.A. I had previously a B.A with a minor in NT Greek from a Bible college. Then I went to the U of M (Minnesota) and got a major in classical Greek (and minor in Latin). I had courses reading Homer (Odyssey 1-3), Euripides (Medea), Aristophanes (Clouds, Wasps), Herodotus. I was reading the NT and LXX at the time off to the side. I was managing myself (no supervisor), at that point I took Greek prose composition.
I really did not feel adequate in composition. I had been trained in Koine, and after all that classical reading, was still an Attic beginner. I could read somewhat OK, but I had no composition skills at all. (My beginning Greek course used Sumner, and we did have to compose the English sentences, but that was the extent of my experience in composing.) At the U, we used Dennison's Greek Prose Style composition book. It was an overview, not something like Sedgwick. After that course, I never did any more composition at all...until 20 years later.
20 years later, after a hiatus, I got back into Greek. I quickly got back to the level I had been. (I did not even remember what a genitive absolute was - incredible.) Slowly I got involved in B-Greek. Then one day Randall Buth asked me to answer him in Greek -- totally intimidating and frightful. It took me about two hours to compose even the simplest of sentences to answer him. My answer was probably 10 words in length. So why was that? I could read the NT easy enough at that point, but I could not compose even one sentence in a quick response. At that point I started listening to Buth's audio and reading Epictetus and recording Greek passages.
I also started wanting to sing songs in Greek -- worship songs and hymns, and so I started composing - and then checking my composition (my phrases) against the phrases of the LXX and other Greek writings. Slowly I began to be able to write Greek. At the same time I began teaching a beginning Greek class, maybe 25% spoken Greek the second year -- but some Greek was better than no Greek. The next year, 50% spoken, and this year 80% spoken. After reading a lot of books on 2nd language acquisition, I'm more convinced than ever that a class needs to be 95% spoken in the second language.
Several years ago, myself and a couple of other Koine enthusiasts started Schole (ning.schole.com). Schole is an attempt to mirror those Latin enthusiasts at Ning.schola.com. We are not as purist as Schola, and allow comments in English. But what happens at Schole is that students of Koine start to communicate in Greek by writing or speaking. It may take several weeks or months to feel the least bit of confidence in composing, but the point, is that a person is trying to USE the language of the NT. Language was meant to be heard and spoken, not just read -- the written language always came second to the spoken and heard.
So I see the biggest questions in regard to composition as the following:
1) When should a student of ancient Greek begin to start composing?
2) How valuable are English to Greek exercises for the beginner?
3) Should a student start freestyle (as on Schole) or start in a formal program (e.g. using Sedgwick)?
4) How much reading experience is needed before a person starts to compose?
5) Is it possible to compose GOOD GREEK only knowing how to read the NT?
6) Does someone who wants to compose need a tutor who monitors his every written phrase?