I think the fact it says their pieces is kinda saying you can't move your opponents
In which case it's at least ambiguous. I take "their" as a plural possessive pronoun and (feminist lobby permitting) "his" as the singular. So "their pieces" in the article would mean only the pieces belonging to the players (plural).
Had FIFE intended to restrict each player's moves to his own pieces they would presumably have worded it, "The game of chess is played between two opponents each of whom moves his pieces on a square board called a ‘chessboard’.
Did you just assume the genders of the players?
No. When I was at grammar school the masculine form was taken to embrace the feminine.
A knight that has movement dimensions 7x3