Only Actions Speak Reaction: After an action is announced: Negate all Honor gains until after the action's resolution. At that time if this negated any gains, negate all Honor gains until the end of this phase and draw a card.
It's my turn. I use Oyo Seido or any other honor gaining card in the limited phase. From what I understand OAS, via a Reaction, creates an ongoing effect.
http://rules.l5r.com/CE_Timing#Timing_sequence
So we use C from the above to decide when the reactions are played.
C. Reactions may now be taken following an action round procedure. Starting with the active player, each player in turn order has the option to take a Reaction to one of the triggering occurrences or pass. When all players pass consecutively, no more Reactions to any of the triggering occurrences may be taken.
As the active player I could pass on a reaction, my opponent then plays Only Actions Speak, and then I could play Only Actions Speak as well. (As a reaction to announcing the action from Oyo Seido). At this point, no honor has been gained or negated.
We then continue through Oyo Seido's action. When I would normally gain honor, it will be negated. But whose Only Actions Speak negates it? As this determines who draws a card, yes?
Does it happen in the order that the reactions were played (in this case my opponent's OAS), or because both OAS's set up effects that happen later, the active player (me) determines whose negates the honor gain. (Choosing my own OAS, so I draw the card.)
At the same point, second question. If one OAS negates the honor gain, is there any honor gain left for the second to also negate?