Saruman and the Blue Wizards ventured into the east, and Alatar and Pallando never returned. Gandalf's remarks about Bombadil imply a long-standing relationship and knowledge of Tom.
Gandalf refuses possession of the Ring in order to avoid its corruptive power; Saruman is obsessed with it, and tries to make his own. These two, whom I presume to be 'upper scale' Maiar, are affected by the Ring, so one would expect Alatar and Pallando to be affected as well. Tom is not affected by it.
If you're interested, Tolkien did offer some speculation about the fate of the Blue Wizards. I've compiled those writings
here in our
Middle-earth FAQ.