I don't think they ever lose their sense of identity (as, for example, the Borg do) and they're clearly not a hive mind.
How could we tell? The Link never spits out two distinct individuals, unless one of them is Odo. All we ever see is a generic Changeling, regardless of whether it takes the shape of Female Changeling, Admiral Leyton, Chief O'Brien or a seagull. There is no known variation in personality or identity. Except for Odo and the other independently operating youngster Laas.