I thought that actions only appeared on the client where they were added?
So basically, all you have to do is only add an action to the player on his machine.
Note that init fields are run on every client, so if that is where you are adding the action, then that would be the problem.
You could solve this by adding the action in the init.sqs, which gets run individually on each client:
player addaction "whatever.sqs"
Since this line of code would only be run ONCE on each client, and 'player' would refer to a different unit on each client, then it should work. On a dedicated server the server might sport an error, since it doesn't have a player, but I don't think it would be a problem.
Anyway, let me know if this doesn't work.