This is a tough one to fully describe with words, and even more difficult to get right without supervision. But I'll give you some ideas that might satisfy your curiosity, and leave it to a hands-on instructor to put it to use.
The lady needs to be aware of both her left and right sides, however, each has a different "role". The left side is picked up and projected forward internally, whereas the right side is picked up to the man.
To get the feeling of the internal pick-up of the left side, sit in a chair to immobilize the lower half of your body. With your fingertips on your sternum and elbows stretched outward, rotate your top to the left, then return to neutral, then rotate to the right, then return to neutral.
While twisting to one side or the other, take the forehead in the opposite direction. For example, when the sternum is rotated to the right, the left side of your back is picked up and forward, which gives your head the support it needs to stretch leftward and backward. You can work on stretching this position by keeping the fingers of just the right hand on your sternum whil placing the fingers of your left hand on your forehead, and pulling them in opposite directions. (Of course, you will need to switch hands when rotating to the opposite side).
The twist to the right (head left) represents closed position, as well as outside partner on right. The twist to the left (lead right) represents promenade position, as well as outside partner on left.
Remember, however, that the forward projection of your left side is an internal position. That is to say, it is entirely a matter of how the parts of your body are positioned in relationship to each other, not to anything outside of the body (such as a partner). You could, for example, have your back to the man and still have your left side picked up and projected forward within your own body. In other words, how you relate to the man does not need to affect how you position yourself internally.
This is an important distinction, because most lady students who first learn how to project the left side forward tend to do so with the body as a whole, literally rotating around the man. This takes the lady's right side away from the man, diminishing the most crucial point of contact. This might be your teacher's complaint.
Once you've established your internal position, then your job while walking into position with the man is to give him your right side. If you have a tendencey to "go around the corner", as it were, then you need to take the position of your body as a whole and turn it slightly to the left, giving him your right side. Within your body, your left side will still be picked up, but your right side will now be connected to the man.
I hope this explanation makes sense. As I said, it's difficult to put into words without any visual aid. But perhaps I've at least answered your question as to how both sides play into the lady's position. Throughout your dance career, sometimes teachers will pick on your left side, and other times your right. Which side they pick on depends on which is lacking -- The internal (projection and pick up of left side) or the external (pick up and connection of right side/rib to man).
Regards,
Jonathan Atkinson
www.ballroomdancers.com