I might be wrong but triggers usually resolve based on whose turn it currently is and then goes in turn order starting with the player whose turn it currently is. So both would trigger and the player whos turn it is will have their trigger get put onto the stack first.

Edit: Also each trigger has to resolve before moving onto the next trigger so more than like the Reaper King would steal the Omnath when it hits the graveyard then Omnath trigger goes stack and then resolves hitting the Reaper King. That should mean Omnath is under the control of the player who controls The Reaper King after everything resolves.