That’s a good question and one that I often find is not well understood even by some scientists working with tDCS or tES generally. I’ve been meaning to do a blog post on it for ages.
The short answer to your question is that current is carried into and out of the brain via free ions. Positive ions at the anode flow through the skin into the brain and build up along the axons and around the cell body of neurons. This excites the cells under the anode. The reverse happens at the cathode where positively charged ions leave the brain resulting in an inhibitory effect on the cells under it.
In its natural state, the brain is full of positively and negatively charged ions. So what I’ve described is a net effect. There may be exchange of both positive and negative ions in both directions.
Ions permeate the skin through a number of channels. Paths of least resistance may include hair follicles, sweat ducts or other secretary glands. In any case, skin is naturally porous and its conductivity will itself increase as it becomes more saturated by the electrolyte medium and even as you sweat.
Normal brain activity involves the build up of positively charged ions around the outside of neurons that eventually results in their activation (firing) and subsequent repolarisation. Introducing additional charge from transcranial electrical stimulation simply augments this process. The overall amount of current introduced is actually small relative to what is already there.
You’ll often see images of tDCS that seem to indicate that current flows through the brain and round in a circuit; but actually, such images are somewhat misleading in my opinion.
I’ll try and elaborate on all this in a blog post.
To answer your final questions: the overall exchange of ions is absolutely related to the current flow that can be measured in the wires of the driving circuitry; the montage that is used absolutely has an effect on which bits of the brain are excited or inhibited and may have some effect on the current flow, depending on resistance; and all of that makes a big difference to the outcome — so in some way or another, it all matters.