The permission is needed for moving through, i.e leaving the hex. You can always enter with or without permission.
Careful with your phrasing here. If you start your movement in a hex with another player's noble, you don't need his permission to leave it. It's only if you are both
moving into an enemy controlled hex and
moving out of it with the same order. If you are only moving in with this order, or only moving out, there's no permission needed.
- If you move into an area that contains 2+ other enemys you have to ask permission to all other: you are red and you move to an area with a green and a blue, if you want to attack blue you need green's permission.
Another clarification: you only need permission from another player to move through or to attack a city (p. 16 of the rules), not to start an open-field battle. You don't need permission from anybody to start an open field battle.