There have been many good answers on this subject. Having been an observer of the missionary process to learn a language, I will share my insight.
They do immerse themselves into the language from day one in their training. As much as possible, they try to avoid their native language and attempt to communicate in the new language as exclusively as they can. This indeed accelerates the learning process.
However, even accelerated, even the military has studied the LDS language training process and has attempted to duplicate it as much as possible. Even they cannot seem to match the success rate from what I have seen, but they try.
When a missionary goes out, they often are not very fluent in the new languages. This makes for some very comical situations when they use a wrong combination of words that suggests something far from what they intended it to mean.
On the spiritual side though, there have been countless witnesses of missionaries who, when fairly new in the language, barely able to speak it well, if at all, have experienced moments when their minds just opened up and marvelous words seemed to flow from their mouths for the duration of a time, stunning not only themselves but those around em.
I believe very firmly that the gift of tongues is given to these missionaries. I believe that anybody who is sincere in their hearts in Jesus Christ can also experience this wonderful gift from time to time. After hearing the witnesses, there is no question in my mind that this gift is manifest in people far more than any of us realize.