How many pieces of paper are there?

Your question seems to be rather scientific than spiritual which reminds me the "Observer effect" in Quantum mechanics. Since this is a forum about Buddhism, we should find an answer for your question from the doctrine.

The author of the article discussed a deep concept of emptiness in simplified words. In the article it has mentioned that how Ven. Thích Nhất Hạnh explains the emptiness in a different perspective, by seeing fullness in every dhamma. The author may not be aware of the questions you've raised but rather added the quoted part to emphasize the idea of fullness which can be seen in the paper.

Your conclusion,

There are two distinct, co-existing pieces of paper. Thoughts?

may be correct according to the Sarvāstivāda school of buddhism. Sarvāstivādin hold the idea of all exists (sarvām asti). According to Sarvāstivāda, there are unlimited number of co-existing papers of past, present and future.

If I answer your questions from a Theravāda perspectve:

Prior to my perception of the paper, was it the same sheet of paper?

The answer is NO.

By seeing my answer you may ask,

Then, was it another paper prior to my perception?

The answer for that question is also NO.

Let me explain this. There's no paper we can find in reality. Paper is a conventional thing which is known to our consciousness. In ultimate truth, we can only find 8 basic elements of matter (Rūpa) in the paper. There's no such thing called "paper" other than these 8 rūpa (Pathavi, Apo, Tejo, Vayo, Vanna, Gandha, Rasa, Oja). These are called "Suddhastaka".

These bundle of rūpa arise (uppada), exist (titi), and then cease (bhanga) rapidly. Then new bundle of rūpa similar to previous bundle of rūpa arise, exist, and cease at the same place or nearby. Speaking conventionally, we can say paper arise, exist, and cease rapidly. By considering the continuance we can say, the new paper belongs to the same stream of rūpa. But it's not the same paper which has already ceased.

Note: This is what I understood. I may be wrong but not Dhamma.

The article introduces three types of dependencies.

The first is causal dependency:

Without a cloud, there will be no rain; without rain, the trees cannot grow; and without trees, we cannot make paper. The cloud is essential for the paper to exist. If the cloud is not here, the sheet of paper cannot be here either. We can say that the cloud and the paper inter-are. “Interbeing” is a word that is not in the dictionary yet, but if we combine the prefix “inter-” with the verb “to be,” we have a new verb, “inter-be.”

If we look into this sheet of paper even more deeply, we can see the sunshine in it. If the sunshine is not there, the forest cannot grow. In fact, nothing can grow. Even we cannot grow without sunshine. So we know that the sunshine is also in this sheet of paper. The paper and the sunshine inter-are. And if we continue to look, we can see the logger who cut the tree and brought it to the mill to be transformed into paper. And we see the wheat. We know that the logger cannot exist without his daily bread, and therefore the wheat that became his bread is also in this sheet of paper. And the logger’s father and mother are in it too. When we look in this way, we see that without all of these things, this sheet of paper cannot exist.

The second is compositional dependency:

In our bodies we have lungs, heart, kidneys, stomach, and blood. None of these can exist independently. They can only coexist with the others. Your lungs and your blood are two things, but neither can exist separately. The lungs take in air and enrich the blood, and, in turn, the blood nourishes the lungs. Without the blood, the lungs cannot be alive, and without the lungs, the blood cannot be cleansed. Lungs and blood inter-are. The same is true with kidneys and blood, kidneys and stomach, lungs and heart, blood and heart, and so on.

The third is conceptual dependency:

Looking even more deeply, we can see we are in it too. This is not difficult to see, because when we look at a sheet of paper, the sheet of paper is part of our perception.

The concept of the paper in our minds depends on our perception of the paper through our senses. In this way, the mental concept of the paper depends on the perception of the paper.

Next, the article introduces the Mahayana concept of emptiness and self:

If we ask, “Empty of what?” he has to answer. And this is what he said: “They are empty of a separate self.” That means none of these five rivers can exist by itself alone. Each of the five rivers has to be made by the other four. It has to coexist; it has to inter-be with all the others.

In our bodies we have lungs, heart, kidneys, stomach, and blood. None of these can exist independently. They can only coexist with the others. Your lungs and your blood are two things, but neither can exist separately. The lungs take in air and enrich the blood, and, in turn, the blood nourishes the lungs. Without the blood, the lungs cannot be alive, and without the lungs, the blood cannot be cleansed. Lungs and blood inter-are. The same is true with kidneys and blood, kidneys and stomach, lungs and heart, blood and heart, and so on.

When Avalokita says that our sheet of paper is empty, he means it is empty of a separate, independent existence. It cannot just be by itself. It has to inter-be with the sunshine, the cloud, the forest, the logger, the mind, and everything else. It is empty of a separate self.

The Mahayana self refers to the self of things, not self of persons. That means, the separate existence of a thing. And the Mahayana emptiness means everything is empty of a self, meaning everything is empty of a separate existence of a thing. And why is this the case? This is because everything is subject to the three types of dependencies - causal, compositional and conceptual dependencies. Nothing can exist independent of each other.

How did things come to have a self? That is, how did things come to have separate existence? How did a single piece of paper become objectified and classified into a single piece of paper?

That happened in our mind. It's called reification or objectification-classification or papanca in Pali or prapanca in Sanskrit. We objectify and classify things relative to our self i.e. into things which are self and non-self, things which are related or unrelated to our self, things which are a threat or which profit ourselves etc. A single piece of paper is a single piece of paper because a human observer said so.

If a tiny ant walks on a piece of paper, it might classify the paper differently compared to how a human does it. A snail might look at a piece of paper and classify it as a type of food it can eat.

So, how many pieces of paper are there? Well, that depends on you.

You might also find this question interesting, where I try to link Mahayana emptiness to Theravada emptiness, through papanca.

I guess according to your definition, the "two" are not "co-existing" -- because one is "before" and the other is "after", so they're not at the same time.

I think that, conventionally, people would say that there's one piece of paper -- no matter how many people are thinking about it, nor whether anyone is looking at it and when.

The article says something poetic, about the cloud existing in the paper. Perhaps that implies that the paper-on-the-table and the paper-in-the mind are the same piece of paper too ... or that they "inter-are".

The bit of the article that stands out for me, is ...

If we ask, “Empty of what?” he has to answer. And this is what he said: “They are empty of a separate self.” That means none of these five rivers can exist by itself alone.

I think that the anatta doctrine is something like, "There isn't a 'myself' that's independent of other things, e.g. of the shandhas."

And Thich Nhat Hanh explains that the sunyata doctrine means that, for each skandha, "There isn't an 'itself' that's independent of other skandhas." (I assume that means that pereption depends on form and vice versa, and so on).

Also interesting to note that the paper is composed entirely of non-paper elements (e.g. minerals, human labour, sunlight).

Anyway, he's saying what the sutta means when it says that "form is empty".

Also, which is less clear, "because of emptiness, form exists" (I say "less clear" but perhaps that's a classic example of taoism -- also it quotes Nagarjuna's saying, "Thanks to emptiness, everything is possible", so I suppose Nagarjuna explains that too somehow).

The bit at he end of the article, about the leaf and the tree, is quite nice too: reassuring, tender.

He seems to be taking the opposite strategy towards self-identification than I'd expect -- not saying "I am not" but saying, "I am everything -- mother, cloud, leaf".

May he be well. :-)