Welcome to PF.
No, it does not work that way. When you shake the first charge, the standard view is that the magnetic field is generated in the charge and starts spreading out to space with the speed of light in every direction. Technically, the field is called to be "retarded", since the field at some distance will be function of past motion of the charge.
Only the first charge has non-zero field, the second charge, being at rest, keeps its electrostatic field.
The magnetic field is present in the sense that moving charges would experience magnetic force. However, as the magnetic force acts only on moving charges, there is no action of magnetic field on the second charge. This means that in the situation as you described it, the magnetic field does no exhibit itself.
However, as the first charge was shaken, besides magnetic field, also electric field experiences change and this will affect the force acting on the second charge.
Where the magnetic field comes from? Based on theory and experience, we picture the electromagnetic fields everywhere and connected to the particles. It is difficult to find deeper explanation for their existence.
Do not try to understand this in terms of some light particles - although the idea is attractive at first, you will find that it is very hard to make it work for every aspect of EM field. It is much better, in your position as a beginner, to try to understand electromagnetic phenomena with classical theory of electromagnetism.