As a supplement to this question, as to whether particles can be observers, let us suppose that the answer is yes. One could suppose a setup where particle $A$ is observing particle $B$, but what is to stop us switching viewpoints around here and supposing particle $B$ is observing particle $A$?
I find this is an intriguing possibility considering the importance of symmetry in Physics.
Question: Is there a symmetry of observed & observer in QM?
Particles don't observe each other, they simply interact according to the Hamiltonian (if they didn't quantum mechanics wouldn't be able to explain the hydrogen molecule, for instance).
If you have two macroscopic observers, and they try to observe each other they would both collapse individually (though their collective wave function might remain the same, depending on interpretation).
However, this effect is absolutely unnoticeable because most states are experimentally indistinguishable from thermal equilibrium (look into quantum equilibration to understand more on this).
So: yes, symmetry is preserved.
I guess this sort of question depends heavily on the interpretation of QM you are "believing" in. However, IMO, measurement or observing is a undirected relation. Consider a two-dim. system with orthogonal states $| 0\rangle$ and $| 1\rangle$. The von Neumann "pre-measurement" will establish entanglement between you, the experimentalist observing the system, and the system itself: $(|0\rangle|\text{you measure 0}\rangle+|1\rangle|\text{you measure 1}\rangle)/\sqrt{2}$.
The Copenhagen interpretation would suggest that the state collapses into one of the two terms. In relative state interpretations, each term would correspond to a distinct reality or "world". But regardless what interpretation you choose, the measurement process is symmetric and it's meaningless to ask whether you measure the system or the system measures you. Note however, that I'm not aware to all interpretations of QM.
Oops. Read only the title, so answered the question, if an observer can observe themselves. The actual question is meaningless because if one person observes the other, then the one who is observed is not the observer.
Answer to the question I (wrongly) inferred from the title:
Short answer: no.
Full answer: in a system that properly includes the observer, there are states that the observer cannot distinguish in principle. Thomas Breuer called it "subjective decoherence".
So, the observer can try to observe a system where he is properly included, but he will fail determining its quantum state. In other words, the quantum state of such system does not exist. The wavefunction is not well defined.
