Different net external forces If I apply an external force to an object, it gets transmitted internally and cancels out according to Newton's 3rd law. This leaves a net external force, causing the object to accelerate. 
But when I attach two objects of different masses by an inextensible string and pull one object with a force, say 50 N, why does it not get transmitted throughout equally, like the internal forces, but instead shares itself between the two objects, say 40 N and 10 N, such that the acceleration is the same?
 A: Realistically, the redistribution of the force between the two masses is the same idea as the force being transmitted internally.
Within a single object, each individual "particle" will receive an appropriate force to cause it to undergo an acceleration that is equal to the rest of the particles within the object. Particle here referring to small parts of the object, be that molecules or atoms, the effect is the same.
As the two masses are joined by a string, so too must the total force be transmitted to all the particles in the second mass such that the entire system undergoes the same acceleration. The 10N and 40N are just the totals of the forces on each particle in each separate block, in just the same was as 50N is the total of the entire system.
On another note, if the acceleration of two masses were not the same, then either the string would need to extend, which you have stated is not possible, or the string would lose all tension and become slack, preventing one mass from accelerating until the slack was taken up and the tension restored. 
