The following is a very small smaple, no more than half a dozen people. But it is based on my personal observations, so it may be of some value:
There are some international students at my school (I live in Canada), who have attained a decent level in English. I have sometimes asked them, how they became so good at English. Some gave very vague answers; perhaps they felt my question was impolite, which is fair enough. But of the ones who were willing to to discuss their language-learning history, it was interesting that their stories were so similar. IIRC, their home countries included Brazil, Iran, South Korea, and perhaps a few others.
To begin, they went to decent schools, but nothing out of the ordinary. English was a compulsory subject, taught with the usual grammar and vocabulary lists to memorize. Not surprisingly, this alone did not enable them to do very much with the language. But around the ages of 12 to 14, their parents starting sending them to extra English classes, outside of school. These were usually held for a few hours a week, usually on a Saturday. (They still had other school subjects to study, so they could not spend too much extra time on English.) The classes were mainly conversational, with only occasional forays into grammar instruction. The emphasis was on learning to use the language.
Note, though, that grammar was not ignored, or considered useless. They would teach it if needed. But they could generally assume that their students were already learning English grammar in their regular school classes. So the instructors could concentrate more on applying that knowledge to conversations.
This went on, Saturday after Saturday, week after week, though sometimes with some holiday breaks, usually until they were 18 or so, and went off to university. So, depending on the age they started, this could be 4, 5, or even 6 years of weekly conversational classes in English. By the end of this, the students found that yes, they could hold reasonable conversations in English, including with native speakers.
I suppose that as they improved, they could make use of more native language materials, especially on the internet. Though they did not mention this.
Also, regarding the cost, they said it was not that bad. It was something a middle-class family could afford. And indeed, it does not sound so very different from the extra classes and lessons that a North American family might arrange for their children. The difference is that in Canada and the United States, they are more likely to be for such things as the piano, figure-skating, gymnastics, a martial art, and so on. (Though to be fair, there are some places that offer heritage language classes for children on the weekend.) Parents in these other countries instead chose to send their children to extra English classes, that's all.
What about those students who did not really want to say how they learned English? It is hard to say. Perhaps some of them did learn much, simply by watching movies and television series. But I am a little doubtful. In my observation, those who feel that they did it "all by themselves" (whether or not that is really accurate), are usually quite willing to tell this to others. (One need only look at the posts made by some members to this forum
) My suspicion is that, at least for some, they did not wish to sound as if they were "privileged" in having these extra classes, and so did not mention it. (Even though, as mentioned above, these classes were not all that expensive for many people.)
At any rate, for the ones who were willing to discuss it, it seemed clear that these extra conversational classes (or at least, several years worth of them), were the key in allowing these students to convert their regular school studies into a real, useful ability to use English. But again, to say that either component could be dispensed with does not seem to be true. Both were essential. And in any case, both were a matter of conscious, directed study. There was nothing random about any of it.
Just my observations.