The picture of two electrons throwing virtual photons at each other is not meant to be taken literally. The Feynmann diagrams you've undoubtably seen are just a graphical representation of a calculation not a literal exchange of a particle. To get some background on this read Matt Strassler's excellent article on virtual particles.
The force between the electrons is indeed mediated by the photon quantum field, but it's not literally an exchange of virtual photons. An electron is an excitation in the electron quantum field. This excitation is continually interacting and exchanging energy with the photon quantum field to create excitations in the photon field (Matt Strassler's article refers to these as ripples). In the absence of any other electrons this energy exchange must average out to zero to conserve energy, though temporary deviations away from zero are allowed by the uncertainty principle. If you bring up a second electron then the interactions of the first electron with the photon field can reach the second electron, and vice versa, and this allows the two electrons to interact with each other.
The transfer of energy in the photon field can be modelled as (an infinite sum of) virtual photons, but it's very different to the excitation of the photon field that we call a real photon. For example virtual photons can travel faster than light.