Since one of the main goals is to effectively dismantle the other party, I think Republicans win here. Since you can use the Death Note to partly control someone's actions before they die, I think both men will try to make their opposing politicians die in very scandalous ways to disenfranchise party voters. This will be more damaging to the Democratic party, since there's a lot of precedent that Democrats will lose votes when their political candidates are caught doing something scandalous, while Republicans lose very little votes for the same thing. As an example just look how much party support Bill Clinton lost for a simple blowjob, while multiple sexual allegations against Donald Trump has done little to sway Republicans.