Newton concluded that the force of attraction between two massive bodies was . also interact with the original pair through their mutual gravitational interactions.

Satish,

Yes.

But the effect is small - only deliberately designed craft (able to sense these tiny accelerations) and which fly sufficiently close to each other, can demonstrate this.

https://www.researchgate.net/publication/256935392_Compensation_of_gravitational_attraction_disturbance_to_pure_gravity_orbit_for_Inner_Formation_Flying_System

Generally, satellite mission designers do not need to consider this force.

I think if you take a look at the gravitational force of an artificial satellite in comparison to Earth, you will quickly realize the huge difference. Just pick Newton's universal law of gravitation and put in the numbers.