>which would result in the piece of paper experiencing a force with magnitude of  2Ff in the opposite direciton of  2Ff.

It doesn't. I remember having this confusion too.  Remember about internal forces or reactive/supporting forces for the entire system . Don't neglect drawing a free body diagram of the paper itself.

When you press a massless block against a table it doesn't experience double your applied from the table pushing back to support it. Imagine a horizontal force sensor on each wheel to measure the normal force: Intuitively (I hope) you don't add those numbers up, and they always read equal to each other.

**UPDATE:**

>I realized the mistake I've made. I'm just confused as to how the net force on the paper is zero, yet its velocity increases (depending on the angular velocity of the wheel). There should be some work done on it and I struggle to understand where exactly it comes from.

Therein lies the distinction. Free body diagrams are a model to equate force to acceleration which is why they are drawn according to F=ma.

So if you're interested in the acceleration of the paper then you need a FBD of the paper at some point. If you only draw FBDs of the wheels you are going to miss something.

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When a free-body diagram says the net force along that axis is zero which means it does not accelerate. It does not mean the pressure/force experienced internally by the body is zero (internal forces), because that's not what you set up to
5 mins ago
 
The normal force applied to the paper in the horizontal direction by cancels out to zero so the paper has no horizontal acceleration. But each roller also has its own friction equation and in the FBD for the paper, that equation appears twice (once on each side), and this equation only considers the normal force from a single wheel for which there is no net zero. That's why you need the FBD for the paper.