Elementary Particle of Magnetic Field
The gauge boson associated with the magnetic field is the photon.
Electric and magnetic fields are in effect different views of the same thing, i.e. the electromagnetic field, and the gauge boson for the electromagnetic field is of course the photon.
Consider you are looking a static charge, which obviously has just a static electric field. But now suppose I am moving relative to that charge. This means the charge is moving relative to me, and a moving charge generates a magnetic field. So you see an electric field generated by the charge while I see a magnetic field. That's why I say electric and magnetic fields are just different views of the same thing.
Footnote: I see Lupus Liber has added an answer that goes into more detail about how the electric and magnetic fields are different views of the EM field, and I recommend reading his answer though you may find it hard going. You might also be interested to read the answers to Do photons truly exist in a physical sense or are they just a useful concept like $i = \sqrt{-1}$?.
There is a particle mediating the electromagnetic interaction: the photon. In the quantum version of electromagnetism (which is a particular example of a quantum field theory), the existence of mediator boson particles for forces is implied.
The following may be worth mentioning:
We say "electromagnetic" (and not "electric" or "magnetic") because this is the only meaningful Lorentz invariant label for this interaction: Magnetic and electric fields are only blocks in the electromagnetic field strength four-tensor, and therefore, are "rotated" into each other under Lorentz transformations. This means that statements like "${\bf E}\neq0$ and ${\bf B}=0$" are true only in a particular frame of reference. This is a classical physics statement, which holds even before quantizing.
By "photon" we mean a quantized plane wave mode of the field. Such a mode has a definite four-momentum. Note, however, that it does not correspond to the case of a classical field configuration. Such a configuration can be constructed by superposition of plane waves and of different multiplicities of modes, which corresponds in the quantum theory to states with various number of photons and various momenta.
It is useful to redefine what a photon is by subtracting the expectation value of the field in the ground state (which is called "vacuum"), given the boundary conditions. This "vacuum expectation value" (VEV) part will automatically obey the classical (in the electromagnetism case: Maxwell's) equations of motion, and the newly defined photons are the quantized fluctuations on top of the VEV.
Detecting photons: yes, this is indeed implied, though practically difficult to measure. For example, a $e^+e^-$ pair may be produced. The pair may emit photons. Since each such emission has a low probability, due to the weakness of the electromagnetic interaction ($\alpha\approx 1/137$), such quantum effects occur with very small probabilities, but they are certainly predicted by quantum electrodynamics. I don't know about the experimental status of this.
About 40 years ago there was an intensive search for the magnetic monopole or magneton. If it were found, the theory of the electromagnetic field would become substantially more complex. However, the magnetic monopole was never observed and the theory remained unchanged, as described in the answer of John Rennie.
This article explains the details:
https://en.wikipedia.org/wiki/Magnetic_monopole
Nevertheless, there have been some claims of the discovery of the magnetic monopole. These results have not been reproduced by others and therefore not accepted by the scientific consensus:
https://piers.org/piersproceedings/download.php?file=cGllcnMyMDA5TW9zY293fDVQM18xODc5LnBkZnwwOTAyMTkwOTI1NTI=&usg=AFQjCNFskwO9f0QA02RMDujVLfsNg5B_XQ
Today, the name of "magneton" is used to describe physical constants of magnetic moment along with other concepts thus creating ambiguity:
https://en.wikipedia.org/wiki/Magneton