How is the energy for a single mass defined in a system composed of two masses connected by an ideal spring? If the total energy is the sum of the kinetic energies of both particles plus the potential elastic energy and the potential energy is "shared" between the particles, if one had to tell the energy of a single mass, could one say that it only has kinetic energy and that the additional energy is in the bond between the two?
 A: It's intuitive but wrong to look at the potential energy as 'belonging' to this particle or that particle. We should really consider the potential energy of the system (two particles plus spring).
Consider an analogous situation. We lift a mass $m$ to a (modest) height $h$ above the Earth's surface. Common wisdom has it that the mass has now acquired potential energy:
$$\Delta U=mgh,$$
where $g$ is the Earth's acceleration.
In reality it's the sytem Earth-mass that acquires potential energy. The illusion that it's only the mass $m$ that acquires the potential energy, arises from the observation that if we release that mass, it will fall and acquire kinetic energy:
$$\Delta U=\Delta K$$
$$mgh=\frac12 mv^2$$
The truth is that even the Earth moves a little, only imperceptibly little because of its huge mass.
In the case of an extended mass-spring-mass system with comparable masses one can clearly see how potential energy is converted to kinetic energy of both masses, as both will move towards one another on release.
