We don't know. Tolkien himself vacillated on it.
In Unfinished Tales, Tolkien says:
[of] the Blue little was known in the west [of Middle-earth], and they had no names save Ithryn Luin 'the Blue Wizards'; for they traveled to the east with Curun'r, but they never returned; and whether they remaine in the East, pursuing there the purposes for which they sent; or perished; or as some hold were ensnared by Sauron and became his servants it is not now known. But none of these chances were impossible to be."
Christopher Tolkien mentions in Unfinished Tales another addendum by his father:
Indeed, of all the Istari, only one remained faithful, and he was the last-comer [Gandalf].
Later, in Letters, Tolkien says:
I really do not know anything clearly about the other two - since they do not concern the history of the N.W. I think that 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.
We also have, from Peoples of Middle Earth,
“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 Romestamo - 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 the East … who would both in the Second and Third Age otherwise have … outnumbered the West.”
So really, we have three possibilities:
- They were ensnared by Sauron.
- They fell and founded magic cults.
- They succeeded insofar as they weakened Sauron's allies in the East.
But, per Tolkien, we don't know which of these possibilities, or combination of possibilities, actually occurred for a fact. We don't know exactly what their task was, but concerning your point regarding Aragorn and Rhun, it seems clear by the time of the War of the Ring, many thousand years after the Blue Wizards arrived, they had not convinced all the peoples of the East to abandon Sauron - and that's probably a lot to expect. They could have easily kept Sauron's eastern allies in check or in chaos for hundreds or thousands of years and we simply didn't know.