All I've ever been able to find is that they arrived in Middle-earth and left to go do something outside of the scope of the story. Has anyone ever found any additional information?
In a letter written during the writing of The Unfinished Tales, in which Tolkien first named Alatar and Pallando, also called Morinehtar and Rómestámo, he wrote concerning them:
I think they went as emissaries to distant regions, East and South, far out of Númenórean range: missionaries to enemy-occupied lands, as it were. What success they had I do not know; but I fear that they failed, as Saruman did, though doubtless in different ways; and I suspect they were founders or beginners of secret cults and 'magic' traditions that outlasted the fall of Sauron.
However, in The Peoples of Middle-earth, written by Tolkien in the last years of his life, he said that the Blue Wizards had succeeded in turning the tides of conflict in the East during the Second and Third Ages. Whether this is a revision of what he previously wrote, or whether they were initially successful and then fell, is not known.
The quote from The Peoples of Middle-earth is as follows:
The other two are only known to (have) exist(ed) [sic] by Saruman, Gandalf, and Radagast, and Saruman in his wrath mentioning five was letting out a piece of private information.
The 'other two' came much earlier, at the same time probably as Glorfindel, when matters became very dangerous in the Second Age. Glorfindel was sent to aid Elrond and was (though not yet said) pre-eminent in the war in Eriador. But the other two Istari were sent for a different purpose. Morinehtar and Rómestámo. Darkness-slayer and East-helper. Their task was to circumvent Sauron: to bring help to the few tribes of Men that had rebelled from Melkor-worship, to stir up rebellion ... and after his first fall to search out his hiding (in which they failed) and to cause [? dissension and disarray] among the dark East ... They must have had very great influence on the history of the Second Age and Third Age in weakening and disarraying the forces of East ... who would both in the Second Age and Third Age otherwise have ... outnumbered the West.
In MERP, the Middle-earth Role Playing Game, the Blue Wizards were said to have both fallen in ways similar to Saruman. Alatar became addicted to Hunting and the other Blue Wizard, Pallando developed a fascination with death. While they started as friends, in the end they became rivals and may well have killed each other off.
Gandalf, the only one of the Wizards who first refused the mission, was in the end, the only one to stay true to it.
To add to Gabe's answer, there is another work (also written in the final year of Tolkien's life) which mentions the success of the Blue Wizards.
To be more specific, Tolkien says that the interference of the Blue Wizards was the reason that Sauron was not at sufficient strength to be able to resist the Númenor invasion.
Thus it was that though, as soon as [Sauron's] disguise was pierced and he was recognized as an enemy, he exerted all his time and strength to gathering and training armies, it took some ninety years before he felt ready to open war. And he misjudged this, as we see in his final defeat, when the great host of Minastir from Númenor landed in Middle-earth. His gathering of armies had not been unopposed, and his success had been much less than his hope. But this is a matter spoken of in notes on “The Five Wizards”. He had powerful enemies behind his back, the East, and in the Southern lands to which he had not yet given sufficient thought.
The Nature of Middle-earth - "Note on the Delay of Gil-galad and the Númenóreans"
"The Five Wizards" is the name of the work where the passage Gabe quoted from The Peoples of Middle-earth comes from. Both writings date from the final year of Tolkien's life and together paint his final view on the fate of the blue wizards.