In some (many) cases wikidata has a better list of multingual names than OSM; however from what I can see they aren't being mapped onto OSM objects, even when tagged. This is therefore either a bug report or a feature request, I am not sure which.
This would be very useful, Maptiler are doing this with very good results. Selfishly I would like the la.wikipedia.org maps to display more Latin than they currently do (see for instance Villa Santi Albani (St Albans)), especially where articles about towns with Latin names then display these names in English on Wikimedia's maps. I suspect this happens with other language content also.