It is likely that they learned of Sauron's demise and departed for Valinor.

^ No evidence of this.

The only thing that is written is that the went into the east and the south and very little is heard of them after that Mandos probably knows, but he didn't tell us.

@Fandyllic I edited the comment to fix this.

@Fandyllic The original post by KaraZorElisLongPig asked what people believed happened to them. Tolkien may not have written about the ultimate fate of the Blue Wizards, but people can have headcanons and speculate about the ultimate fate of the Blue Wizards.

...and? I just said there is no evidence of that. If people are going with headcanon, they should say that.

I believe that they were killed at some point before completing their mission.

@Fandyllic well ofcourse people are going with headcannons. The question was "what do you believe happened" not what actually happened, since we don't know what happened to them.

Make sure to read the questions. 😉

I think, not sure, but I feel like I've read somewhere that the Blue Wizards were corrupted in some form or led astray by other earthly matters and forgot their original goals? And so they probably just spent the rest of their days wandering in the wilds...or something like that I imagine

Unfinished Tales states that they either died, became lost to Sauron or started cults in the East upon Morgul arts.

So its intentionally left a mystery.

Towards the end of his life, Tolkien returned to the issue of the other two Wizards. In a brief outline he noted that the two Wizards were sent to Middle-earth in the Second Age and were destined to disrupt the work of Sauron in the East:

"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".