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 Numenorean 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 Roleplaying Game the Blue Wizards were said to have both fallen in ways simmilar 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.