1) The codex for reanimate says that the creature is summoned.
2) The FAQ on page 21 states that the mage who controls the effect responsible for the reanimation is the object that performs the summon.
3) The codex for controller says that each spell or object in the game is controlled by the person who casts it.
Therefore, the mage who controls the Rise Again spell gains control of the creature they used it on.
In the Rulebook v2.0 on page 18 it states that when two players wish to reveal their enchantment at the same time, the player with initiative gets to go first.
In the FAQ on page 8 it discusses Duplicate Enchantments. It is not legal to knowingly duplicate an enchantment, so the only time it can happen is when two opponents do it accidentally. When this occurs, whichever enchantment is revealed first becomes the legal enchantment. As long as the revealed enchantment is in play, the duplicate can not legally be revealed. It is allowed to remain in place unrevealed indefinitely, but if its controller attempts to reveal it without removing the first enchantment beforehand, then the second one simply gets destroyed without effect.
Thus, with Rise Again, they would both trigger at the same time. The player with initiative would get to reveal their's first. The creature would be reanimated (placed face down and considered removed from play until the end of the round) and the Rise Again would be discarded. Then the 2nd Rise Again could be revealed. At this point, it's effect is invalid since it can not reanimate a creature that is currently removed from play, so it gets destroyed without effect.
That is my interpretation of the rules.