Before discussing any such matters, we must define our internal force sign conventions. It is common to define them thusly:
So, tension is positive, compression is negative. Shear forces are positive upwards when on the left side of the cut, but downwards on the right side of the cut (a good way to remember is that positive shear rotates the beam clockwise). Bending moment is positive if it generates tension on the bottom of the section.
When an external bending moment is applied to a beam, it resists deformations by generating an internal stress state. This stress state is such that it can be described by an internal bending moment which is equal and opposite to the external bending moment, cancelling it out and allowing the beam to remain at rest in this new deformed configuration. It also generates zero total axial force (under pure bending).
As can be trivially seen from the deflection, your beam is under negative bending moment. This means the top fibers of the beam are elongated and the bottom fibers are shortened, meaning the internal stress state goes from tension at the top to compression at the bottom.
So, if we cut the beam at midspan, we can choose to look at either the left or right side of the cut, which will give us:
It is obvious that both sides must be equivalent, since they represent the exact same things in the exact same spot, only looking at different sides of it. And indeed, they are exactly the same, once we remember the sign convention.
- Both sides show tension (positive) in the top, compression (negative) in the bottom. Good.
- On the left side of the cut, the tension on top and compression in the bottom would generate a clockwise rotation, which is negative. On the right side of the cut, they would generate a counter-clockwise rotation, which is also negative. Good.
- I didn't represent shear in the image because it isn't relevant to the current question, but we'd expect the forces to either side to be equal and opposite.
Therefore, there is no benefit to displaying the stress distribution on both sides. If you have one, it is all you need and the other is entirely redundant. Now, this also means that one is not more correct than the other. So, if you're going to write a diagram, should you do it with positive axial forces going to right (as in the right side of the cut and in the example you gave in your edit) or to the left? It's entirely up to you. It is customary to have positive axial forces going to the right because, well, usually positive values go to the right.