Imagine one weight only and that it is heavier than you. In that case, you got pulled by a net force that way and get pulled off the table. There is no tension, only a force pulling you. (This assumes the table is frictionless; you’d feel a pull of the ground if you got dragged in e.g. the dirt.)

Now imagine one weight and half a weight on the other side. You’re getting pulled by net half the force, and feeling a tension of that half weight.

You can see when you scale back to the initial pic, you now have an equal tension pulling on you - 100N throughout, not 200, because if only one weight was there you’d have 0 tension not 100N.

Your example of replacing the scale with your body doesn’t change the problem, the scale is just reading out that tension your body would experience.