Why aren't charged particles constantly shining?

Well, one thing to be clear. The exchanged particles are virtual particles (in your case the virtual photon). Since they are not real, they shouldn't be detected.One way to know that there are indeed virtual photons is by studying the Feynman propagator: for free particle case, various virtual photons are excited at all directions. If you would like to know the influence of this at some other point, you do integration to get the probability. In fact the particle does excite virtual photons at all time. But these are not the photons in our everyday life, with, which you can indeed detect them. We are affected by these virtual photons but the method to analyze this process is through the method called QED(of cause, you know it).

Charged particles exchange virtual photons. These are not the same as usual photons in electromagnetic spectrum, so they can't be detected directly. However, virtual photons "sea" can be detected indirectly, because quantum vacuum energy density is solely due to quantum vacuum fluctuations, aka. virtual photons. And these virtual photons are the main cause of Casimir force,- attraction or repulsion between close parallel plates. This force arises, because virtual photons makes a standing waves between those two plates. No any other serious explanation of Casimir force exist, so this force can be considered as a good proof to existence of virtual photons.