I think your perplexity is understandable, and it comes from the clash between the notion of stress, which belongs to continuum mechanics, and the molecular description. Let me arrive in a roundabout but hopefully insightful way at why the factor of 2 is unnecessary.
The notion of stress (more precisely: internal pressure) was introduced by Euler around 1753 and then generalized by Cauchy around 1828. Euler's question, summarizing a little, was the following: if I have a body of matter, delimited by some boundary, how can I represent the total force exerted on it by the matter outside of it? His idea was to consider forces that act purely on the boundary (just like when we have something pressing on our skin). The total force on the body of matter could then be found integrating this field of surface forces over the whole surface. His revolutionary idea was that we could imagine to delimit an arbitrary inner portion of a body by an imaginary surface, and consider the forces acting on such surface. Euler considered only forces orthogonal to the surface, and Cauchy generalized them to forces with arbitrary directions – for example tangential to the surface: that's what viscosity is. Cauchy also showed that such a force $\pmb{t}$ could actually be expressed by the action of a linear operator – the stress tensor $\pmb{\tau}$ – on the normal to the surface: $\pmb{t} = \pmb{\tau}\pmb{n}$.
The invention of stress also suggests the best way to think about it, in my opinion. Do not imagine two sides of a surface. Instead, imagine a 3D portion of matter delimited by a closed surface. The stress is just a field of force which that portion of matter "feels" on its surface, caused by external agents. The stress is called "tensile" if the force is directed outwards and pulls on the surface; it's called "compressive" if the force is directed inwards and presses on the surface.
In the case of the rope, imagine a portion of it, even if very short, delimited by two circular surfaces: a short cylinder. You want to know the total force exerted on this 3D portion of rope from the rest of the rope (or anything else outside). The lateral surface has no forces acting on it. Each circular surface does have a force acting on it – the stress – directed outwards with respect to our short piece of rope, which thus feels a pull at its extremities.
In continuum mechanics stress stands in contrast to so-called "body forces" or "volume forces", which instead act on every small volume of a body of matter. Chief example is gravity. Thus body forces $\pmb{f}$ scale like a volume, while stresses $\pmb{t}$ scale like an area. The total force on a body of matter $B$ is then given by the contribution of both:
$$\pmb{F}_B = \iiint_{\text{bulk of $B$}} \pmb{f}\ \mathrm{d}V + \iint_{\text{boundary of $B$}} \pmb{t} \ \mathrm{d}A \ .$$
The centre of mass of the body will move as if this total force is applied directly to it.
You see from these ideas and equation that there's no need for a factor $2$.
The question of the "two sides" of the surface appears when there is a body of matter $B_2$ adjacent to the first $B_1$, so that they partly share a delimiting surface $S$. By Newton's third law, if $B_2$ is pulling on $B_1$ at the surface $S$, then $B_1$ is pulling $B_2$ at the surface $S$, in the opposite direction. So if you consider the surface $S$ from $B_1$'s perspective, the stress is directed outwards, towards $B_2$. And if you consider the surface $S$ from $B_2$'s perspective, the stress is also directed outwards, towards $B_1$. The situation is no different from when we say that the Earth pulls on the Moon, and the Moon pulls on the Earth with an equal and opposite force. Only, in the case of surface forces this pulling is happening on the same spot. That's what's often confusing. But the two forces are acting on different bodies – keep this in mind.
At a molecular level surface forces don't exist. All forces are body/volume forces. The notion of stress doesn't apply here in its original sense. What we consider as stress on an (imaginary) surface from a macroscopic point of view, turns out to be one of two things, or a combination of both, from a microscopic point of view.
First: atomic/molecular body forces having short range: just few layers of molecules on one side of the imaginary surface act on just few layers of molecules on the other side. That's why, from a macroscopic perspective, we consider these forces as only existing on the surface itself.
Second: motion of molecules across the surface. Since the molecules carry momentum, momentum is decreasing on one side of the surface and increasing on the other. And since a force causes a change of momentum, macroscopically we interpret the microscopic change of momentum as a force existing on the surface. The decrease of momentum on one side of the surface is equal and opposite to the increase on the other; so the macroscopic intepretation is that the material on one side is macroscopically experiencing a given surface force, and the one on the other side an equal and opposite surface force. Many viscous forces are of this kind.
A curious final note. In contrast with molecular or particle dynamics, stress is the only kind of force that appears in general relativity instead, because action at a distance is forbidden there. In fact, when we write the Einstein equations in a "Newtonian" form, split into space and time, the Newtonian stress tensor $\pmb{\tau}$ (and the energy, but no momentum or energy flux) fully appears in the evolution equation for the metric.
Euler's article is really cool to read:
A good book to get acquainted with the notion of stress and also its microscopic interpretation is
There's a beautiful lecture by Truesdell on the history of the concept of stress, which can be very useful for its understanding:
The microscopic interpretation of stress was approached in a rigorous manner I believe first by Irving & Kirkwood at the end of the 1940s, followed by many others. Recent reviews are given by
even if the maths in these may be somewhat advanced, from the text and the equations you can get a glimpse of all sorts of different microscopic stuff that contribute to what we macroscopically call "stress".
For the role of stress in general relativity see for example