This slippy map suggests that the osm-multilingual tile server on Labs supports localized maps the same way tile.openstreetmap.de does (example). It loads tiles such as the following:

http://a.tiles.wmflabs.org/osm-multilingual/de,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/en,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/fr,_/4/3/6.png

However, all these tiles contain the same local-language labels, corresponding to the name tag in OSM rather than tags like name:en or name:de. The same is true for any other language selection:

http://a.tiles.wmflabs.org/osm-multilingual/el,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/fi,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/it,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/ja,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/ru,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/sv,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/vi,_/4/3/6.png
http://a.tiles.wmflabs.org/osm-multilingual/zh,_/4/3/6.png

If I omit the _, which is supposed to be the local fallback, I always get a 404 error no matter how many times I retry:

http://a.tiles.wmflabs.org/osm-multilingual/de/4/3/6.png

The old Toolserver tile server worked for all the Wikimedia languages. I had to switch the Vietnamese Wikipedia’s OSM gadget to use tile.openstreetmap.de and would like to hook it up to WMF Labs once its language support is built out.