In the small town of Terni (Italy), there's a couple of young friends named Marco and Leonardo, who like to perform magic tricks to a restricted audience of common friends and relatives. They like to call themselves "The prodigious Duo". Every week-end, The prodigious Duo meets at the house of the leader, which happens to be Marco, to practice new magic tricks, learn new techniques, and, most importantly, to set up the magic show of the week, which will be shown to the audience in the late evening.
Recently, the leader of The prodigious Duo discovered what he claims to be "the coolest magic trick ever done using a chessboard" (which actually is a checkerboard, but that's basically the same thing). Marco only tested the trick on a $2 \times 2$ squares checkerboard, nonetheless he's certain that the trick would work on any checkerboard with size $n \times n$, where $n$ is a perfect power of two (meaning that the binary logarithm $\log_{2}n$ of $n$ is an integer). So he managed to create a working strategy for a real checkerboard (with $n=8$). Once tested the strategy for the bigger board on a piece of paper, Marco is now ready to explain the trick to Leonardo.
He starts explaining the peculiarities of the checkerboard involved in the trick, since that it isn't a normal checkerboard:
"The checkerboard is composed by $8 \times 8$ squares, like a normal one, except that the squares are all red colored, and distinguishable because of the vertical and horizontal lines which define them. This red color choice is only made to avoid confusion when moving the pieces."
"The pieces of the checkerboard are normal coin-shaped pieces, except that they're white on one side and black on the other one."
"The checkerboard is always visible to all the members of the audience during the show, and so will be me and you, remember it, this is important."
Then he explains the magical trick from the spectators' point of view:
"At the beginning of the show, the checkerboard is filled with sixty-four pieces, each one placed inside a different square. The pieces' sides facing up are of mixed color, and their initial color doesn't really matter for the purpose of the trick (e.g. the quantity of black or white pieces doesn't matter)."
"Once the checkerboard is set up, you will ask for a member of the audience (we'll call him $P$) to come up on the stage to participate in the trick."
"Once on the stage, I will cover my eyes and ears with a special mask which will make me unable to see or hear anything around me. Now you'll ask $P$ to rearrange the colors that he sees on the checkerboard flipping any number of pieces he wants, any number of times."
"Now you will take a minute or less to look closely and meticulously at the checkerboard's configuration, and then ask $P$ to: choose a piece among the 64 pieces on the board, flip it only one time, and memorize its position."
"Once $P$ has done its simple task, you'll do the same thing: you'll carefully choose a piece (which doesn't necessarily need to be different from the one chosen by $P$), and flip it one time. At this point, you will lock your head into the same kind of mask I used before, and, once done, ask $P$ to touch me to let me know you cannot see or hear anything."
"Felt the touch and removed my mask, I'll reach the checkerboard, carefully observe it for a bit, and finally remove a single piece from it: the piece that has been chosen by $P$."
"I'll now help you to remove your mask and bow down with you to the surprisingly cheering audience and the astonished $P$, which is trying to work out what sort of sorcery has been going on moments before."
Leonardo takes a minute to elaborate all the information provided to him by Marco, and then, excited, replies: "This is an incredible magic trick, but man, I mean, sixty-four pieces... how could we ever even think about that? Tell me you've got a solving strategy which is as beautiful as this impressive trick!". Marco starts now to explain the real strategy which they are going to apply to recognize the piece chosen by the volunteer on the stage...
You're asked to discover and explain the strategy which The prodigious Duo is going to apply to find the correct piece chosen by the volunteer.
Clarifications and notes
Now, stating some basic facts that might be misunderstood:
During their show, none of the members of The prodigious Duo can communicate with each other, because of the masks they use to cover their heads.
This means that Marco cannot, in any way, see or listen to what's happening on the stage when the volunteer is mixing up the colors flipping the pieces on the checkerboard.
And also that Leonardo cannot, in any way, see or listen to what's going on when Marco removes his mask and looks at the checkerboard, revealing the piece chosen by the volunteer.
No one knows, except for the two magicians, the strategy applied to recognize the piece moved by the volunteer.
The volunteer called on the stage isn't, in any way, collaborating with the magicians nor cheating; same goes for the rest of the audience.
Avoid stupid answers like "Marco checks for the fingerprint leaved by the volunteer on the chosen piece" or similar.
Personal notes
I want to clarify that I didn't think up this puzzle by myself, I only made up some little story-line; I did listen to it years ago talking with some friends of mine, and it randomly came back in my mind recently. For this reason, I do not know the exact solution of this puzzle (and this is the main reason why I am asking for a solution here on SE), nonetheless I'm still able to provide additional details and explanations to anyone who asks for them, editing the question itself or answering the comments. Given that I do not know the real source of this puzzle, it would also be nice if someone could provide it. Oh, and in case you were wondering: yes, that "Marco" is me, ahah.