The short answer to your question is yes. When you focus on the person and then move the camera to recompose, the focal distance remains as it was with a manual-focus rangefinder.
The exception would be if YOU move or your SUBJECT moves between the time you focus and the time you trip the shutter. Then the subject might or might not still be in focus, depending on something called 'depth of field' (DOF).
Focus is about distance, in the simplest form. When you focus on a person X feet or meters away, and then recompose, the focus is still set to X distance. If you move or they move, focus is still set to X, but the subject may not be X distance away any more.
As others have noted, this can be complicated by the fact that focus is a subject that has many characteristics. When you focus at a certain distance, objects in front of or in back of that distance may be in focus or not, depending on this like your lens focal length, the size of your aperture, and other things.
I hope you find this helpful.