It is my understanding that when two magnets collide without physically touching, kinetic energy is conserved and the "collision" is elastic. My question is, what mechanisms allow for this exchange in kinetic energy?
The electromagnetic field itself carries energy. Conservation of energy demands that the total energy of the system, which is the sum of the kinetic energies of the magnets and the energy stored in the EM field, remains constant. This is true for any conservative field theory.
As another example, imagine a system of massive particles initially at rest. The particles begin to gravitate toward one another. So a system which initially had no kinetic energy whatsoever suddenly acquires kinetic energy as the particles begin to move. Where did this kinetic energy come from? The gravitational field. The total energy of the system, however, remains constant.
And to answer your question: how is this energy transfered? By interacting with the field. The field influences the particles and the particles influence the field.
The force between the magnets is the mechanism for exchange of momentum and kinetic energy. The origin and nature of the force does not make any difference. It is the same for all action-at-a-distance forces like electrostatic and gravitational forces, as well as the the more obvious 'contact' forces such as normal reaction, friction and drag.
Momentum is always conserved. When the force is strong enough to cause deformation of the objects, then kinetic energy is lost. Usually the magnetic materials themselves are much stronger than the magnetic forces they create, so they do not deform during the collision.