Actually both sides can be used on the headers I have. The side with the pillars are a bit harder to push but then seem also harder to remove, which is a good thing I presume.
Well, just as long as you never intend to plug anything
else into the socket, as you will most likely have distorted the spring contacts.
So in fact, if you have tried both sides in the socket, you may have
already loosened the contacts to the extent that it will be less secure when you use the side with the thinner pins.
As an aside, it would probably not be a good idea in any case to rely on a single row of pins to mount a PCB. An additional standoff or a second row of pins - such as the Arduino shields use - is certainly more appropriate.