What is measured by electroencephalograms and local field potentials?
[I give this as a sketchy answer being aware that it might be nonsense. So feel free to downvote in this case. I should have mentioned - thanks to Bryan for having remembered me - that this answer has been inspired by Buzsáki/Anastassiou/Koch's article on The origin of extracellular fields and currents and just tries to sketch the mental image this article evoked in me.]

I assume that what is actually measured by an EEG electrode is an effective dipole moment in the upper cortical layers underneath the electrode (which decays with $1/r^2$). The way this dipole moment is created by "brain currents" is the matter of this answer.

There seem to be some premises under which a measurable EEG signal (= a dipole) can be detected:


The presynaptic action potentials must arrive highly synchronized. (Additive superposition.).
The apic dendrites of the pyramidal neurons that give rise to the dipole must be vertically oriented (pointing to the skull, resp. the electrode). (Orientation of the dipole.)
The sodium potassium pumps trying to restore the rest potential are unevenly distributed, i.e. mainly located at the soma. (Breaking the symmetry.)


This is the situation for the neuron (gray) at rest:



The dipole then forms as follows in three steps:

Step 1: An presynaptic action potential releases neurotransmitters that open ligand-gated ion channels causing sodium ions to enter the neuron.



Step 2: The extra sodium ions wander quickly to the soma. This happens passively, i.e. driven by repulsion and diffusion. At the end of step 2, all ions inside the neuron are distributed approximately evenly (i.e. not giving rise to an inner dipole).



Step 3: At the soma sodium ions are pumped out of the neuron. During this step equidistribution of ions is maintained inside the neuron (still no dipole inside the neuron).



In the extracellular medium, we now have an observable dipole.

Summary: What is measured by an EEG are not directly "brain currents" but an effective dipole moment that results from three different currents on different time scales: 


Ions passing the membrane of apic dendrites into the neuron. (Slow.)
A net flow of ions wandering from dendrite to soma (driven by repulsion and diffusion). (Fast.)
Ions leaving the soma, e.g. by sodium pottasium pumps. (Slow.)