A frequent example used when saying that the definition of 'word' is fluid in Japanese particles, which can be viewed as separate words or affixes. I want to know whether a native speaker would say that, for example, 私は was one word or two.
This is a simple and casual question, but it doesn't mean the answer would be likewise. Sometimes a very basic notion in a language turns out to be completely absent in another. One of my favorite examples is "Framing a question whose answer is an ordinal number" on English.SE.
So, conclusion first, if you ask this to native speakers, I'm afraid answers would vary according to personal opinions, because all we can do is to find counterparts from foreign languages we know (that have solid notion of "word") before we decide yes and no. It's very similar to how we sometimes casually say 行った is 過去形 (past tense) or 先生たち is 複数形 (plural), where the analogy goes pretty promising at first, but eventually fail at some point.
The difficulty around the question is twofold:
Word has little practical use in Japanese
In word-dividing languages we must have knowledge about what's word and what's not, otherwise we can't write correctly. Naturally we come to have a grasp of "word-ness" even when we don't write. But in Japanese, we don't put spaces in sentence, don't count by word for amount of text, thus don't pay attention to how many words are in a phrase. I guess most people don't have slight thought about it until they're asked.
Just to be sure, I'm not saying that native speakers cannot perceive noun and particle as separate "units". Blackboard is easily divided into two elements, but it's still one word, at least orthographically.
Japanese particle is neither standalone word nor affix
One of presuppositions behind the question is that, if something isn't a standalone word (linguists call it free morpheme), it must be an affix (bound morpheme). This law of excluded middle doesn't hold in Japanese. What we call particle (助詞) in Japanese is established as clitic, which is in short, what means like word but sounds like affix. It's a relatively new term, but it doesn't mean that it's a creation by linguists to obfusticate people. Rather, it's something hadn't had name due to its marginal presence in European languages, whereas it abounds in some other parts of world. Japanese language has "real" words and affixes besides particles, so particles are, after all, particles.
How clitics are processed in space-inserting orthography is complicated and usually tied with etymology: no space (Latin populusque), always spaced (Polish zaloguj się), hyphened (Portuguese conjuga-me), apostrophed (English I'll), or affected by other factors (French Je te vois / Je t'aime). You can see there's no decisive answer whether clitics should fall under word or affix, from the standpoint of grammar as well as writing (well, because they're neither...).
To answer your question, 私は has two words but other similar expression might not.
I think a part of your confusion comes from what affix means. In the most basic words, an affix changes meaning of the word to which it is attached. Consider these example in addition to 私は. 私に、私へ、私が、私を、私と, and etc....Do you see that some of what follows 私 could be easily translated to "to" or "with"? I believe they are words, though that is not to say all of those can be consideted word. While the meaning of the phrase changes (from subject to object, In a away), the meaning of 私 remains unchanged so these are not affix. Yet, there is no direct "word" to describe を or が so they really aren't word in the strictest sense.
You might argue that there is a difference between I and me, which is the difference between some of the examples above. But remember, we dont have a concept of that. The above example shows how we use the same word to mean I or me.