The most obvious method of learning chess tactics is simply to solve a large number of problems, tackling each one once and only once. Solving a 1,000 problems 10 times is clearly going to be more effective than solving them once, but this is not a fair comparison. How do the two methods compare if you spend the same time on both? Is it better to solve 1,000 problems 10 times, or as many fresh problems as you solve in the same time? The relevant test here is the ability to solve fresh problems. The rate of improvement at solving a fixed set of problems will be greater than that for solving fresh problems - but this improvement will not be fully reflected in an improvement in solving fresh problems.
The motivation for the once through method is the belief that you get good at what you practice. In a real game, you do not have benefit of having seen the positions before, and perhaps replicating this in your training will increase its effectiveness. A disadvantage of the once through method is that there are likely to be long time intervals between occurrences of all but the most common tactical patterns, so learning to spot them is likely be a slow process - and may not happen at all - if you have completely forgotten the previous occurrences. Since memory cannot be used so efficiently in the learning process with this method, the average time spent tackling each problem will be greater than if problems are repeated. You are not going to be able to solve 10,000 problems in the time it takes to solve 1,000 problems 10 times.
The once through method is rather like learning French by watching French films with subtitles, watching each film once and only once. It is very similar to the way that small kids learn languages, but it is not going to be an efficient way for adults to learn. If I had to learn French this way, I would want the films on DVD, so that I could study each section repeatedly, and revise what I had learned.
Although learning chess tactics shares some important characteristics with learning a foreign language, there are also some important differences. The most obvious difference is that chess is visual, whereas language is verbal. There are many reasons why you may fail to solve a tactical problem, but the most relevant are:
* Failure to spot a tactical pattern (e.g. a double attack).
* Failure to study the position adequately before attempting a solution.
* Failure to work out the idea behind the combination.
* Failure to analyse efficiently (e.g. spending to much time analysing one of your less good moves, or failure to consider a defensive move for the opposition).
* Failure to visualise the position correctly during analysis (e.g. failure to spot that a file has been opened).
Repetition can potentially help you address all of these failings (and remember the lessons that you have learned). However, if you take this technique too far, you risk ingraining thought processes that are different from those that you need for solving fresh problems, and in a real game. Your optimum tactics training depends on the capabilities that you have at the start of the training, and one training method may not be enough.
To conduct a fully scientific comparison of the effectiveness of the two approaches, we would need a controlled trial with a large number of volunteers, in which they all abstain from all other chess activity during the trial (otherwise we would not know what had caused the improvement). We would also need a reliable method of measuring the volunteers problem solving capability (on fresh problems) at the beginning and end of the trial. To make sense of the results, we would need some way of profiling the capabilities of the volunteers. For example, if the volunteers all started off with an uncommonly poor knowledge of tactical patterns - that might skew the results in favour of repetition.
Last but not least there is the question of motivation. Repeatedly solving problems has the disadvantage that it is less interesting than solving fresh problems, which does not help motivation. On the other hand, the higher success rate when solving repeatedly aids motivation, so perhaps we have a draw on motivation.