As Scyan explained pretty much, the one who starts banning is the player who rolled (!roll) higher points.
E: Oops, i got trigged even myself, the "Players can only pick two (2) Maps from the same pool" rule doesn't work like that, and actually the only part where it applies at all is the finals (WBFinal, LBFinal and GrandFinal), Since it should read "Each player may pick only two (2) Maps from the same pool". Thus meaning that you may pick Swing it, Clumsy and your opponent can pick another NoMod pick.
TL;DR Players can only pick two (2) Maps from the same pool, Only applies Per Player not per Match!
E: Re-wrote the rule to be easier to understand correctly!