This amounts to a confusion over notation: "proper acceleration" (e.g. as measured by an accelerometer) vs "coordinate acceleration" (the acceleration an observer observes an object to be undergoing).

The acceleration an observer sees you undergoing is the same as the inherent "proper" acceleration you're undergoing, minus the acceleration of their coordinate frame with respect to yours. For me to stay still with respect to you, if you're in a frame that is accelerating away from me, I need some proper acceleration to catch up and counteract the fact that our frames are diverging. But if spacetime is curved, your frame probably is accelerating relative to mine - c.f. the example on the Earth's surface, where our frames inexorably accelerate towards each other as we move parallel to each other. So for me to stay still with respect to you, I need to have some proper acceleration to balance out the coordinate acceleration derived from the fact that our frames are moving in a curved space.