Does writing improve reading comprehension?

[I think this is more properly a comment, but I'm making it an answer because it's too long, yet supplements Ben Kovitz's excellent contribution.]

Over the years I've discussed this very topic with members of a Latin reading group, which includes teachers of Latin. All agree that, to beginners, the compressive effect of 'doing into Latin' from English is hard to accept : the biggest problem from the outset is to persuade students to be less word-by-word literal in composition than instinct tells them. The difficulty is intensified by having to learn a quite alien system of grammar in order to accomplish it.

I don't know of any study on it, but for what it's worth my own experience is that translating from one language to another deepens the understanding of both. Complete mastery of the language requires fluency in both reading and writing Latin : you can't have the one without the other.

I don't know of any studies specifically about this, but I can give you two reasons for expecting writing to help.

1. Here is a classic psychological experiment from the days of behaviorism.* One group of rats, the "active rats", was allowed to explore a maze and find food in the usual fashion—learning by reinforcement. Another group, the "passive rats", was put in wire cages and pulled through the maze along the exact same paths as the active rats. Then both groups of rats were tested on how well they could navigate the maze. The purpose of this experiment was to test the behaviorist thesis that learning can only happen under reinforcement. And indeed it showed that "latent learning" happens: the passive rats did learn the maze. But in the absence of cues from outside the maze (e.g. when the room was painted black), the passive rats did not learn the maze as well. The active rats learned the maze better than the passive rats.
   
   You don't really need carefully controlled scientific experiments to see what's going on here. The same kind of observation is ubiquitous in common experience. If you're driving a car, for example, you can probably learn a route in one or two iterations, whereas the passenger will usually barely learn the route at all. Drawing a picture of someone's face makes you see their face much better than photographing it or just coming to recognize that person by their face. People tend to learn more richly from active behavior, such as speaking and writing, than from relatively passive behavior such as reading, though of course we learn from both, and of course there's no simple rule. A passive participant is usually less attentive to the cues that inform decisions because the passive participant is under no pressure to attend to them. An active participant cannot escape the pressure to focus attention on the most relevant cues.
   
   

2. When you go through the process of framing and expressing your thoughts in a language, you become better at understanding the language. You gain the ability to empathize with the author of speech or text that you hear or read, because now you've gone through that process yourself. Understanding language is mainly learning to empathize with the speaker: "What is the speaker trying to point out? Why is the speaker choosing these words or this sequence or this grammar to help a listener see what he is trying to point out?" Even in your own native language, writing makes you a better reader than reading alone.
   
   To put this another way, language is communication. If you only practice receiving, you miss out on the insights that arise from sending—and consequently you're less prepared to infer a sender's intended message. When you put your own thoughts into language, you cannot avoid attending to such matters as what knowledge you are assuming that your listener already has, whereas it's easy for reader to read obliviously to these things—especially a reader who doesn't know the language's conventions for conveying them.†
   
   

The ultimate test, of course, is to try it yourself. Case studies and anecdotal reports from other teachers might save you a lot of trial and error on your own, as long as they include rich descriptions of what happened rather than only the author's conclusions. Maybe another answer can provide those.

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* McNamara, H. J., Long, J. B., & Wike, E. L. (1956). "Learning without response under two conditions of external cues." Journal of Comparative and Physiological Psychology, 49(5), 477–480.

† See the introduction to Latin Word Order by Devine and Stephens (2006).