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.
Well, I know the literature isn't canon, but I've always had an interpretation of the Great Link which matches how they were later portrayed in the novels.
Even in the show, Odo and the Female Changeling have both come and gone from the Great Link many times, and they always remained independent entities.