Can I reverse geocoding from my local OSM server?

Installing your own Nominatim instance with a separate copy of the data is the best way to do it. However if you only use it occasionally and you don't mind a small performance penalty you can also hack together a few PostGIS queries that will give you the nearest street, city and so on from a normal rendering database, for example to find the nearest street to a point with the coordinates mylat,mylon you would do something like

SELECT highway,name
FROM planet_osm_line
WHERE highway IS NOT NULL
AND ST_DWITHIN(way, ST_MAKEPOINT(mylon, mylat), 300)
ORDER BY ST_DISTANCE(way, ST_MAKEPOINT(mylon, mylat))
LIMIT 1

or to find the city boundary that contains the point,

SELECT name
FROM planet_osm_polygon
WHERE boundary='administrative'
AND ST_CONTAINS(way, ST_MAKEPOINT(mylon, mylat))

You might have to wrap the ST_MAKEPOINT in a ST_SETSRID(..., 3857) or ST_SETSRID(..., 900913) depending on your setup.