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When the right atom gets a push, the bond between the left and right atom gets slightly stretched. This induces the tension force pulling the atoms back together as I have indicated. Since the left atom is initially at rest, it will move in this direction.
I will edit the answer above with an sketch of the system at different time steps. At each time step I will draw the force due to the bonds also the current velocity of each atom and I will also draw the new velocity vector. Recall that the force is proportional to the change in velocity. For example, $\vec{v}_{t+\Delta t}=\vec{v}_{t}+\vec{F} \Delta t/m$
If the bond behaves like a spring (which is acceptable for small oscillations), then the bond only exerts forces along the line connecting the two atoms. The misunderstanding here is which inertial frame of reference you are dealing with. In the centre of mass frame your masses will rotate as you have indicated. If you are in the rest from where the centre of mass is moving up, both atoms will be moving upwards, however they will have a relative velocity for example, the atom on the right might have v_cm+v, while the atom on the left has v_cm-v.
