Sometimes it seems as a submissive that we are supposed to feel guilty if we have desires of our own. The idea of us selflessly giving and serving is the ideal. If you wish for things beyond the privilege of being allowed to submit... it's often seen as bad. That makes you a "do me" sub. etc.
All in all that is part of the game. There are enough terrible sub-wannabes out there that make Dommes jaded and bring about this hard-line stance.
I'm not going to really debate the merits of the game. As I've stated in the past I believe that submission provides "something more" for a sub... usually some mix of emotional and sexual fulfillment that makes submission worth doing.
Fairy tales are full of sympathetic protagonists, living destitute lives of abuse and labor and carrying hopes and dreams of a better life. It is only the submissive that longs for the reverse... they are usually in a functional life with too much freedom. It is the submissive that longs to be taken from their free life and driven back to the beginning of the fairy tale.
Blurred lines upon lines... convoluted ideas... where does the boundary between submission and "do me" fall?
When you look at the sexuality of a submissive I think it is important to remember that subs tend to be heavily fueled by the psychological side of things. While they will surely respond to direct physical stimulation, they are also very likely to respond to dynamics and environment. If we consider an erection (or attempted erection) to be the most basic form of arousal, I believe the answer to the question may fall in with the depth of arousal.
To look at the extreme end first, I believe the tipping point for a sub happens when they reach the state you could describe as "horny." At this point they have been aroused to the point where they desire physical stimulus in order to ejaculate. Their brain no longer functions rationally as they seek physical gratification, despite their beliefs or best intentions. I classify this state as overstimulated.
It is in this overstimulated state where a sub can become "do me."
I tend to strongly equate the depth of my own subspace with my level of arousal. I must be aroused to reach subspace, and "sufficiently" aroused to reach deep subspace. This becomes a bit trickier as it relies heavily on the idea of sexual frustration. The sweet spot is then to be stimulated enough to where the chemicals and hormones allow for my psychological submission but not stimulated enough to where I become horny.
Thus it becomes a game of tease & denial, keeping a sub aroused enough to feel submissive but never permitting enough stimulation to push them into their base mental state.
I do have to say that a Domme that is able to keep their sub off-balance and wanting in such a way truly has a fearsome skill.