The Greek text for John 1:18 says:
θεὸν οὐδεὶς ἑώρακεν πώποτε ὁ μονογενὴς υἱός, ὁ ὢν εἰς τὸν κόλπον τοῦ
πατρὸς ἐκεῖνος ἐξηγήσατο (TR)
Let's break that down, word-by-word, in order to help understand it.
||has made [Him] known
Translation from the Greek
As you look at each of the words in the table above, which are listed down in the same order they appear in the Greek, you will see that in places they appear to be switched around as compared to an English translation. That is because Greek words have grammatical forms, called declensions, which indicate the role of the word in the sentence. English is always SVO (subject-verb-object) in basic sentence structure and order; but Greek can put the object first, or place the words in a different order, because each word indicates whether or not it is the subject, the object, etc. Verbs and adjectives also indicate the same declensions, and can be tied to the noun or pronoun for which they are applied. Verbs, like in Latin languages, carry the pronoun/person of their subject as well.
It so happens that the first word of this verse actually is an object. "Theon" is the "accusative masculine singular" noun for God, and "accusative" means it is the object of the verb.
The adjective which follows is in "nominative masculine singular" form, with "nominative" applying to the subject. Greek does not need to repeat subjects, or even supply their place with a pronoun, if the subject can already be inferred from a prior mention, etc. In this case, this clause of the verse has no subject, but the verb, together with this adjective, imply the subject as "no one."
When we get to the next noun, notice the sequence of definite article, adjective, and noun, ALL three of which indicate "NMS" (nominative masculine singular). This indicates plainly that these three are all part of the nominative clause--they are together: "the only begotten Son."
This is then followed by another definite article of the same form, but which must be assumed to apply to a new noun. However, that noun is not present. This is because Greek grammar does not require the noun to be repeated if the same one as was just addressed is still in focus. We can understand, therefore, that it is the Son still being referenced. In English, this might be translated as an appositive, or as a dependent clause based on the earlier noun with a word like "who" or "which" so as not to repeat the word "Son" when it has not been explicitly repeated in the Greek.
The word "being" is the verb here. It is "present participle active - nominative masculine singular." Remember, verbs carry their subject pronouns with them, so we know that the subject must be masculine (grammatical gender) and singular--and the word "Son" certainly fits this.
Next comes the object, indicated by the "accusative" words "the" and "bosom."
The following definite article is of a different declension: it is genitive. This means one of several things, but usually it implies possession or belonging and is translated with "of" in English, in addition to the word with which it is associated--in this case the article (the), which is why the translation is rendered as "of the." The genitive noun to which the article corresponds also indicates its grammatical use in the sentence, completing the expression "of the Father."
Finally we have a demonstrative pronoun in nominative masculine singular form, where, again, the "nominative" indicates the word is a subject.
The final verb is in a special Greek form that has no English equivalent. It is the aorist indicative middle voice, third-person singular. While it could apply to either gender, we are already given the masculine gender explicitly in the prior word, so we know that it applies to a he/him. This Greek verb can be translated to a reflexive verb in English (which is why the "him" appears in brackets), even though it's technically not the same thing in Greek. There is no direct translation possible when one language lacks the same grammatical form.
If I were to summarize this translation, and render it in my own words, I would read it as:
No one has yet seen God. The only begotten Son, being in the bosom of the Father, he has revealed Him.
As you can see, the Greek grammar fairly tightly controls how it can be read or interpreted. There is much more degree of detail to its grammar than we would have in English. Even then, ambiguities can sometimes exist, but in this particular passage, it would be difficult to point to any ambiguity of significance.
As is readily apparent, the expression "who is himself God" is NOT in the Greek text here. There is one variant reading of this text, known to be a late manuscript change, in which the word "Son" is changed to "God" (Theos). But even this textual variant cannot account for this added phrase in the NIV or the other versions referenced. For simplicity, I will address only the NIV in what follows.
Why the NIV Adds "who is himself God"
While I cannot speak authoritatively for the translators of the NIV, it is worthy of note that in order to copyright a derivative work based on some prior text, a certain percentage of the content must be changed. That percentage is not quantified, mathematically (for example, the 30% rule is a myth), but it is qualified in terms of being significant or substantial changes or additions.
The NIV was most certainly among the most highly copyright-protected Bibles when it first came out. The publishers have relaxed their stance somewhat since then (current rules for licensing), but, for example, any website wishing to include the NIV originally was required to pay $10,000USD up front, with additional royalties based on visitor counts afterwards. In order to obtain this copyright, substantial changes were required. (I heard they had to change 15% at minimum, but there seems to be no actual numerical figure encoded in the law--perhaps this was just their own goal in order to substantiate their claim to copyright.)
Actual legal instruction for copyrighting a derivative work can be found HERE.
By the way, the current NIV has "who is himself God" in this text. The NIV of 10+ years ago did not have this addition. The NIV copyright rules enforce immediate and unannounced upgrade of their text in all official websites once they make a change (no prior copies are allowed to remain). I happen to have a copy of the NIV text from before this change was made. If you have a print edition from a decade ago, you may be able to verify this.
From a hardcopy of the NIV of a few years back:
No one has ever seen God, but God the One and Only, who is at the
Father's side, has made him known. (John 1:18, NIV)
The front of the Bible has these lines:
Copyright © 1986, 1992, 1996 by The Zondervan Corporation
Copyright © 1973, 1978, 1984 by International Bible Society
If the modern Bibles are evolving, does the truth itself change over time?
The NIV and NLT have no textual support for their "who is himself God" addition. The NIV's own track record stands at odds with its current reading.