There are many reasons why the buildings might be misaligned relative to specific imagery: the original imagery they were traced from might have been not well aligned or the imagery you are comparing too might have a large offset.
I had a quick look at the history (in OSM) of some of the buildings and it seems that they were traced from 50cm resolution imagery from SITN in 2009. I would as a tendency rather trust the SITN imagery than bing in this case.
Anyway there seems to be higher resolution imagery available see http://lists.openstreetmap.ch/pipermail/talk-ch/2012-October/002119.html , however it seems as neither JOSM nor iD has the source in its default configuration. I’ll add that some time today (it typically takes a while till it becomes available though, but if I add it to our proxy server, you will be able to conficure it manually in iD).
Naturally imagery from 2006 is dated, but at least for the old part of the city it should still be useful.
PS: the Swiss mailing list tends to be more active than the forum.
However, it seems like the roads are aligned on Bing and the houses on that other source you mentioned, which is probably not optimal. Which source should I now use when adding new elements? And is there a way to select either of those two groups of elements and align it on the other one?
As I have only been using the on line tool on a quite amateur way, I would not risk tempting anything like that on my own, but right now it’s a bit weird trying to add missing info into this mess. Ideally I would link you to a problematic area, but I don’t know how that works. You can for instance take a look at the “Chemin de la Caille” in Neuchâtel and the surrounding houses. Some of them have only less than 25% in common with the map…