How does EM radiation depend on the reference frame? In special relativity, magnetism is electrostatics in a different reference frame. This is how we explain the magnetic field being produced by moving charges (aka currents). Charges that move produce electromagnetic waves, due to similar effects (the speed of light, propagation of effects etc.).
So, if we have a static charge in a reference frame does it radiate electromagnetic waves in another reference frame that is in relative motion to it?
In other words: is it possible, to place a static electric charge (producing only an electrostatic field) for example on the moon, with no EM radiation being detectable if one is next to it, but being detectable at a long distance by an object that is moving relative to it?
 A: If I understand correctly you are asking how observer dependent is electromagnetic radiation. 
The first thing is that non uniformly accelerated charges are described in a inertial frame by Larmor's formula and Abrahm-Lorentz force which take into account the radiated field and the recoil on the particle.
Now in special relativity and Newtonian mechanics acceleration is an observer independent concept. For example in special relativity acceleration is meant with respect the non dynamic flat spacetime (Minkowski spacetime). In Newtonian mechanics with respect the Newtonian space and time. This last framework is the context where the above description of an accelerating particle applies.  
So if those theories were how the world is described completely the radiation  only depends on the absolute acceleration of the charge which is  independent of the observers.
However, in the real world we know that at least GR and QFT must be consider for a more complete description. Does the picture change when this theories are take into account? 
If we add QFT and allow the background to be curved but still static.  The answer is yes and is given  by the Unruh effect. Observers detect thermal baths depending on their state of motion.
If we add GR but remain with a classical description of Electromagnetism then there is not a definite answer. The problem arises because of several reasons. First we don't know how to solve Einstein's equation with point particle sources. The second is that in GR there is the assumption of the complete physical equivalence of the gravitational field and a corresponding acceleration of the reference system. If the answer to the uniform acceleration is yes, then static charges in a uniform gravitational field must radiate or the equivalence principle is wrong.
Moreover,as pointed in this answer the description of a uniformly accelerated charge particle is not settled even at the level of classical Electromagnetism in Minkowswki spacetime. The difficulty is in knowing how to correctly account for the influence of a charged particle on itself.
