It's usually very easy to maneuver around the un-initiated, the vanilla, world at large. Not always, but usually. And discipline need not be spoken. Most times, all Y/you need is a knowing glance to know what message the O/other is trying to convey. This gets easier the longer Y/you are together and in tune with one A/another.
Nor does punishment need to be physical. I rarely use a physical form of punishment Myself, because I feel that 1) the punishment should always fit the offense, and 2) the punishment should serve as a lasting reminder and a way of teaching or correcting bad behavior or judgment. And I ALWAYS follow this up by requiring an essay (of various lengths, depending on the severity of the infraction) that lets Me know that, not only does she understand what she did wrong, but also WHY it was wrong, what a correct alternative would have been, and how she plans to avoid the incident in the future.
Now, as with everything, this is only My style and every relationship is different. I just share it as the only real-life example that I have a working knowledge of, and perhaps it might be something Y/you find useful in Y/your own life.

D/s vs M/s (TPE):
The only TRUE difference, in My understanding, between a Dominant/submissive relationship and a Master/slave (or Total Power Exchange), is the level of commitment. In a D/s relationship, the submissive gives the gift her submission to the Dominant with every exchange, whether by actual verbalization, or simple unspoken agreement.
In the M/s relationship (TPE), the slave submits to her Master once (as in, once and for always), with total trust and unselfish release of all control of her person, and binds herself to Him for life. This is more than any simple marriage and should not be entered into lightly... but when you know absolutely for certain that He/She is THE One... and allow yourself that FINAL release, the rewards are unlimited.

As always, these are just My opinions, based solely upon My perception and personal experiences. I hope it helps.

PAELUS