Lamor demonstrated that an individual accelerating charged particle emits radiation, i.e. loses energy that is radiated away. So we would say that the fields produced by an accelerating charge have a radiation component.
Maxwell's equations are linear, so when you have a large collection of charges, the total field is the sum of the fields produced by the individual charges. These fields add in amplitude, and so it is possible to have cancellation. It's perfectly possible to have the radiation fields produced by one accelerating charge to be cancelled by the radiation fields produced by the other charges in the collection.
As an example, consider a 2 wire transmission line, we know that there is a solution to Maxwell's equations that is a pure TEM travelling wave along the wires. There is no radiation component - all the energy is directed along the wires. The individual charges on the wires will be moving and accelerating, but their radiation fields MUST cancel in this pure TEM mode. If you isolated the movement of one individual charge, and then removed all the other charges in the system, you would likely find that that moving charge radiated (in the absence of all other charges).
Now, of course, it is impossible to set this up as the TEM fields go off to infinity all around the wires and the wires need to be infinitely long, but in many practical situations this field distribution is a (very) useful approximation, and the deviations from it - including some radiation loss - are minimal.
In antenna and circuit work it is not normally necessary to consider the movement of individual charges, instead to use the current distribution to calculate the cumulative field of all the charges moving in the current.
Added in response to: "Yes okay the radiation fields may cancel out but the energy loss of the electrons will be the same in every part of the circuit "
This is not true, Maxwell's equations are linear in fields quantities, not energy.
Consider a +ve charge moving in simple harmonic motion. It will radiate as it is accelerating, locally you would observe this as a force making it hard to accelerate the particle, and the work you did on the particle to make it follow SHM would equal the radiated energy.
Now consider the same scenario, but this time a -ve charge (equal in magnitude to the first), same outcome. Now place the two charges close together and move them, as a pair, in SHM. In the far field the radiation fields are much smaller as the radiation fields from the two charges tend to cancel, and locally you would find them easier to move as less power is radiated. So cancellation of far fields can allow charged particles to accelerate without (or more probably with minimum) radiation.
If you could just add all the energies, you could radiate EM simply by waving a block of wood around as it contains all those protons and electrons.