In the top diagram each electron emits photons, real or virtual, that go in all directions. The dotted line shows one such interaction where the photon interacts with another photon in another wire. The real photons are from accelerating electrons and these exchange energy with another electron. The virtual photons leave the electron but return after a certain time (a half life). When they interact with another electron the effect is repulsive but when the interact with a proton they attract.
The photons do not obey the inverse square law but the photon density falls off as the inverse square of the distance. The carry vectors of the electrons direction and velocity (momentum) and carry energy as a scalar. They also carry angular momentum. This is imparted to the receiving electron. They travel at the velocity of light.
The effects of current is seen as a consequence of the special theory of relativity.
The total effect of their interactions is the sum of their individual actions.
Forces between coils:
Here is a diagram of the forces between two coils, one inside the other.
As you see there is a force compressing the inner coil and if it is displaced there is a restoring force.