Say, there are 2 stationary electrons placed at a distance. The result of observation would be both flying apart with the same speed and the opposite direction, which would obey laws of conservation of momentum and energy (as electric potential energy is converted into kinetic energy). But if considered from the perspective of emitting and receiving photons, how is the process like? Where does the energy come from? Does the mass of either electron change? Or should I not try to understand it from the classic point of view?
From the classical view there are electric and magnetic fields throughout space.
And they have their own energy. And the energy in the fields changes as the fields change.
And when there are charges the fields change differently there. They change in a way where energy flows from the fields into the charges where the electric field points the same direction as the current and it flows from the charges to the fields in places where the electric field points the opposite direction as the current.
And that's exactly when the kinetic energy of the charges changes. And the energy from the fields is spread out in space where the fields are and it is the energy in the fields right there where the charge is that is exchanged with the charges right there.
No potential energy involved.