What coil dimensions (wire size, coil i.d., center-to-center length, number of turns)? What capacitance?
A DMM will not even respond to anything over a few kHz; you're probably lucky it still works (or... maybe it doesn't).
AC current takes the path of least impedance, which is generally the closest path between two conductors carrying the current out and back. An infinite parallel plate is closest at all points, so the current distributes evenly on both faces. But finite parallel plates (like you get from a mirrored set of nice wide traces on either side of a PCB) have edges. The magnetic field necessarily and implicitly generated by that current, shapes where that current flows. The field is 'pinched' between the plates (because the plates are conductive, which forces magnetic field lines to the outer surface -- skin effect), and where it is able to spread out freely (at the edges), current is concentrated. So, if you have parallel plates, current predominantly flows along the edges, with some fraction of the total flowing down the middle, face to face.
In general, you can observe that current prefers to flow along sharp edges, and facing surfaces. So a square conductor carries more current in the corners than the surface (let alone in the body -- skin effect), whereas a round conductor carries most of the current in the facing side.
Another way of thinking of it: like currents repel (hence, pushing current out into the corners of a square conductor), opposite currents attract (drawing currents closer to the surfaces closer to the return conductor).
So, if you have connectors with round pins, if you use every other connection (per row, or in a checkerboard pattern if it's a multi-row connector), you'll get a reasonable distribution. If you use all the connections on the one side for one polarity and all the others on the right for the return, most of the current will flow through the center pins (where the currents are closest -- path of least impedance).
If you have connectors with flat (blade) pins, a lot of current will flow along the edges, but also along the flat surfaces, so in addition to the generally larger build, the current capacity isn't too bad. As long as the polarity alternates, of course.
Tim