As retained in the Latin Vulgate, and therefore the Douay-Rheims ('the Catholic KJV'), but usually obscured in most English translations, Jesus is actually saying the vast, vast majority of the time, "Amen, Amen I say to you."
Which does mean 'Truly, I tell you a (solemn or sure or serious) truth (i.e. so pay special attention)'.
In Greek, the authors have transliterated "Amen" into Greek (authors who also recorded Him say "I tell you the truth," implying they saw it important to at least record that He did say "Amen" specifically—John records that Jesus in fact said 'Amen' twice in each instance, "Amen, Amen": since he didn't put extra words into Jesus' mouth, the other authors have determined, not erroneously, this to be perhaps lost on their audience, and so retained the term, but not the exact words).
And he said: Amen I say to you, that no prophet is accepted in his own
country. In truth I say to you, there were many widows in the days
of Elias [Elijah] in Israel, when heaven was shut up three years and six
months, when there was a great famine throughout all the earth. And
to none of them was Elias sent, but to Sarepta of Sidon, to a widow
woman. And there were many lepers in Israel in the time of Eliseus
the prophet: and none of them was cleansed but Naaman the Syrian.
And all they in the synagogue, hearing these things, were filled
with anger. And they rose up and thrust him out of the city; and
they brought him to the brow of the hill, whereon their city was
built, that they might cast him down headlong. But he passing
through the midst of them, went his way.
"Amen" is a Hebrew/Aramaic (getting it from Hebrew) word meaing "May it be" or "So be it." Since this is a rather loose translation of the word, which usually followed the end of prayer1 (a faithful assent to the prayer), it means here, in Jesus' use, something close to "Yes, truly, ..." or "Most assuredly, it shall be no other way than..."
The Catholic Encylopedia (Catholic Encylopedia 1914, A, "Amen") says the following:
"So frequent was this Hebrew in the mouth of Our Saviour", observes
the Catechism of the Council of Trent, "that it pleased the Holy Ghost
to have it perpetuated in the Church of God". In point of fact St.
Matthew attributes it to Our Lord twenty-eight times, and St. John in
its doubled form twenty-six times. As regards the etymology, Amen is a
derivative from the Hebrew verb aman "to strengthen" or "Confirm".
So Jesus is highlighting those truths which He deems most important to be remembered or believed, not implying these are true and the rest of what He says is not true—God forbid. Jesus also appears to be the only Person to use "Amen" before what He says or affirms as most true, which is interesting.
1 Neh 8:6; cf. Judith 13:26—this seems to be a double expression of the usual one "Amen!" (1 Chron 16:36), and not the sense in which Jesus is using "Amen." That is to say, 'A solemn or sure truth is about to be spoken:'
Jesus is said to be "the Amen." (ὁ Aμην ho amEn) Jesus is the solemn Truth (John 14:6).
And to the angel of the church of Laodicea, write: These things saith
the Amen, the faithful and true witness, who is the beginning of the
creation of God:
Amen in the context of Jesus is here tied to true/reliable witness (as far as I see it).
My Humble Opinion
When Jesus uses two "Amen"s, I believe He is actually showing that the Father affirms and approves and bears witness (represented by the first "Amen") that all He (represented by the second "Amen") says is most true and reliable. Not out of any necessity (John 8:13-14) but because it's something that's important to know for us who are not Jesus (John 8:14) and need to have witness from the Father:
If I bear witness of myself, my witness is not true.2 There is another
that beareth witness of me; and I know that the witness which he
witnesseth of me is true.
2 That is, true witness—valid witness for all intents and purposes; as far as we are concerned
The fact that St. John's Gospel is focused on and written to highlight the relationship between the Father and the Son, I find this not impossible, and quite probable.
However, in John 16:7 He actually says, "εγω την αληθειαν λεγω υμιν.." which means "I tell you the truth.." but could just as easily be translated, equally so, though it would not be normal for anyone to say this except Jesus, "I, the Truth, tell you.."
However, He is still emphasizing that what He says will certainly come to pass (John 16:4).