Talk softly, carry a big stick, and don't worry about credit

In July 1983, the world took notice when the officially atheistic Union of Soviet Socialist Republics allowed a small group of Pentecostals (who had been living in the basement of the U.S. Embassy since the summer of 1978) to emigrate to the West.  Over the preceding five years:

Many journalists interviewed them while American ministers and U.S. congressmen visited frequently.  The U.S. Senate even passed a bill giving them permanent residency status. But nothing made the Soviet government budge. (Reagan's secret legacy--quiet diplomacy)

Thus, the obvious question was "Why now?  What happened to change the inscrutable Soviet mind?"  One emigre--Liuba Vashchenko--when asked this question, responded, "It is difficult to understand the [Soviet] government or why they do things." (Reagan's secret legacy--quiet diplomacy)

Though no one knew how to explain this outburst of humanity from the Andropov government, that didn't stop speculation.

1. Some [State Department] officials linked the announcement to the East-West conference in Madrid. (Hundreds welcome 16 Russian Pentecostals) 
2. There was speculation that unnamed Embassy officials had simply bartered with their Russian counterparts and--luckily enough--were not double-crossed this time. (Reagan's secret legacy--quiet diplomacy)
3.  Rev. Phillip Potter of the World Council of Churches maintained that, "We (i.e., the WCC) played a not insignificant role in the recent case of the Pentecostals." (Church rights: Restrictions controversial) 
4. Others credited the efforts of an Alabama housewife named Jane Drake and her Society of Americans for Vashchenko Emigration (SAVE), noting she had "traveled across the country and to Russia to rally support..." (Housewife joyous about release)
5. Some writers suggested overtures by congressman Dan Levin (D) of Michigan played a pivotal role. (Housewife joyous about release)
6. Finally, William Beecher of the Boston Globe noted, ""It was only after President Reagan personally intervened…that [the Pentecostals] were allowed to leave.” (Andropov may be very clever chessplayer)

But no one could say for sure which, if any, of these speculations were right.  Over time, additional information has been released demonstrating that Beecher's instincts were more spot on than anyone realized.

In his excellent book, God and Ronald Reagan: A spiritual life, Paul Kengor relates the following story from Reagan Secretary of State, George Shultz:

During [a] dinner [with the President], Shultz commented on the usefulness of his private meetings with Ambassador Dobrynin, which had been opposed by some in the administration...He told the president, Dobrynin would be dropping by his office again...[and asked if Reagan] would like to surprise the ambassador by joining in.  "The president said, 'great,'" remembered Shultz.  "He was itching to engage [the Soviets]."  Shultz figured the meeting would take ten minutes.  It went on for over an hour...In the meeting, Reagan "came down very hard on human rights," particularly the Pentecostals' rights.  He told the ambassador that some positive act by the Soviet leadership might make it easier to resume overall U.S.-USSR negotiations, and suggested that the Pentecostals might serve as that token.  If the Soviet leadership were to take action to resolve the impasse he would be delighted, and would not embarrass the USSR "by undue publicity, by claims of credit for ourselves, or by 'crowing.'"  They cut a deal, the Pentecostals were set free, and Reagan kept his word." (God and Ronald Reagan, 2004, p. 285-286)

The case of the Russian Pentecostals was a perfect illustration of the type of "quiet diplomacy" Reagan espoused in a letter to John Koehler:

“I just don’t happen to think that it’s wise to always stand up and put quotation marks in front of the world what your foreign policy is,” Reagan wrote a friend in 1981. “I am a believer in quiet diplomacy and so far we’ve had several quite triumphant experiences by using that method. The problem is, you can’t talk about it afterward or then you can’t do it again.” (Reagan: A Life in Letters, 2004, p. 375)