If you look at the following location and compare it with both the Satellite Imagery (like DigitalGlobe) or with uploaded GPX traces (via JOSM), you can see that almost all buildings and roads in this area are offset by about 5 to 10 meters:
I would like to (and already did) add/fix a couple of roads/junctions, but if all other roads and buildings have to be moved, it might be better to fix all the locations before actually improving/updating the map.
This is likely to have happened because of poor geo-referencing of older aerial imagery, I’m assuming this isn’t in an active seismic region, otherwise it could really have moved.
I think there is generally a reluctance to fix this sort of error. Instead one adjusts new mapping to the effective local datum.
5m is within the typical tolerance of GPS and if you are going to move things you will need to do an the ground GPS survey, to much higher accuracy, to establish correct calibration of the offsets. With consumer products, you are probably going to have to average over hours to be sure of absolute errors much less than 5m.
I think it rather unlikely that there is a sharp border, within which everything has the same effective offset, so I think you will end up having to blend in the edges in any area that is reasonably well mapped.
One thing that did concern me is that the GPX trails seem to be too accurate. There is an issue, particularly I believe with Apple devices, that the device snaps GPS readings to the nearest road in the device’s internal map. As well as breaching the database rights on that map, it also means that errors in that map’s datum will get reflected into OSM. Before correcting the datum here, I think you should do your own calibration survey with a device known, or configured, not to do that.
On the general principles as to when you should correct a systematic datum error, I think you need to wait for other people to respond.
Since one probably will need to realign the sat imagery to a more accurate coordinate before doing that, I have always wondered how big of an area of imagery that will also be offset. Is it per tile? per downloaded area? per square km?
One typical cause of this is parallax error and it will vary smoothly with height and also be affected by how far off centre the subject is. It is as likely to be related the original aerial photograph, as to the final tile.
Also this image is not a satellite image; it will have been taken from a fairly low flying aircraft.
To answer the question: given that you’ve established that offset is a) real, b) can be corrected. There is no reason to not correct the positions of the objects, outside of it potentially being a lot of work.