There is another possibility which is that in Tolkien's world, the words of a seer have the same power as the words of an oath, i.e. to make a self-fulfilling prophecy.
We know that in Middle-Earth words (and song) have power, power to fulfill each being's innate spiritual power (fëa) in the Unseen World and effect the physical world through the body (hröa).
We also know from the Silmarillion (apart from the references in LOTR and the general medieval literary setting) that oaths have power once taken, as do curses; and that the power of an oath somehow derives from its appeal to Fate, i.e. the Music. When sworn on a powerful creature, the oath-taker is calling upon that being to witness his or her oath. That is why Tolkien suggests (and medieval thinkers decreed) that oaths should rarely be taken, as they have power just like song.
In this context it is possible that prophecy works the same way, i.e. a manifestation of the spirit world upon the physical world through speech, foreordaining a certain conclusion.
People who subscribe to a certain type of magical thinking in real life are very common, to wit, not wanting to make it real by predicting it (i.e. jinxing it). To be a prophet in Middle Earth, one would presumably have to be inspired (i.e. have some glimpse of the Music, through vision or otherwise).
Making false prophecy would presumably carry the same penalty (or curse) as taking oaths in vain. (Note that the curse example can be seen as the penalty exacted by he who to an oath is sworn upon the oath-keeper.)
Supporting Example from the Text (to back up my hypothesis): When Fëanor says "this thing I will not do" in promising not to relinquish the Silmarils, Mandos ominously decrees "So it is doomed".
This could be a prophecy or a promise; either way, when invoked, Mandos accepts it as if it were an oath; in reality, he knows (more or less) what the outcome will be.
Note: while looking up the exact quote, I found this: Green Eggs and Silmarils
Fëanor: I'm Fëanor.
More skilled than any Elf before.
These are my sons: one, two, three, four,
five, six, and seven – maybe more!
I made these gems: one, two, and three.
They are the best. Don't you agree?
You may not hold a shiny gem.
You may not even look at them.
I do not want you at my door.
Can we have your jewels, please?
We need to fix our dying Trees.
I will not give them for the Trees.
Not even if you say "oh please."
You cannot have them here or there (...)