When does Debye shielding occur? I know that the phenomenon of Debye shielding is present when you place an additional charge $Q$ into an otherwise homogeneous, neutral medium. This case is however very special and I was wondering - does Debye shielding occurring in other situations within a plasma and if so when? Does every particle have it's own Debye shield or just a few?
 A: The short answer is that Debye shielding occurs because electric fields do work to get rid of themselves.
A slightly longer answer is that Debye shielding occurs so long as the medium is not dominated by collisions (which act as a drag force limiting the redistribution of charges to a quasi-neutral state) or driven by an external field.
Most plasmas are quasi-neutral precisely because of my short answer above, which is also why large-scale, quasi-static electric fields are rare in natural (i.e., space) plasmas.
A: First lets understand Debye shielding. As you said, we insert a particle of charge $Q$, as our test charge. Then, the electrostatic field generated by the charge will be exponentually attenuated (or shielded) by a Debye length of $\lambda_D$. This shielding occurs because is the plasma itself is made of ionized particles, which will move blocking the field of $Q$. Henceforth, the shielding only depends on the constitutions particles of the plasma. The shielding is independent of the test particle itself.

Does Debye shielding occurring in other situations within a plasma and if so when?

So, Debye shielding occurs everytime you insert any test charge $Q$. By derivation, this is the only situation in which it occurs. In particular, you can consider the test charge, to be any of the charged particles of the ionized plasma with charge $q$, and consider how its $E$-Field will be shielded by other plasma particles. In other words, the electric field of each particle does no penetrate much in the plasma, because the other particles will move to "block" it. Since the necessary condition is only the presence of a charge $Q$, we see its not a phenomenon exclusive to plasma. Anything with zero total charge which has a lot charged constituents, the same will happen. Eg.: Electrolytes, colloids, lots of charged billiard balls floating around. Etc.

Does every particle have it's own Debye shield or just a few?

The shielding is caused not by the test particle per say, but because of the plasma (ie, the other charged particles constituent to your plasma, or electrolyte, or billiard-ball pool) which are blocking the $E$-Field of the test-charge $Q$. This is clearly seen in expression of the Debye length:
$$
\lambda_D = \sqrt{\frac{\epsilon k_B T}{\sum n_j q_j}}
$$
where $q_j$ is the charge of the $j$th particle that constitutes your plasma (or electrolyte, or etc). And $n_j$ is the distribution of the $j$th constituent around the space, assuming there is no test charge $Q$. As you can see, it fully independs on the test charge $Q$.
A: Ok so here is what I have found on this subject. In Fundamentals of Plasma Physics by P.M.Bellan the following passage is given after a derivation of the Debye length:

Finally, it should be realized that any particle could have being "the"test particle and so we conclude that the time-averaged effective potential of any selected particle in the plasma is given by Eq. (1.9). [$\phi(\vec r)=\frac{q_T}{4\pi \varepsilon_0 r}e^{-r/\lambda_D}$]

