When fighting a Rosie, use armor piercing rounds with the machine gun or pistol. Another effective way of killing them is by launching frag grenades and running out a door. The Rosie will follow you, fire proximity mines at the door, and be damaged or destroyed when it walks through it will.
While in levels that have more than one Little Sister in them, there are sometimes Rosies roaming the level alone. The following only happens if you have already harvested or saved at least one but not all the Little Sisters in that level. The lone Rosie will walk up to one of the many holes in the walls through which the girls crawl through and bang on the metal before waiting. Then, it will help a Little Sister jump down from the opening. The girl will then lead the Rosie throughout the level. Therefore, when hunting these formidable enemies, do not attack or ignore a Rosie if it walks alone (as long as in the pause screen it says that there is still at least one Little Sister remaining). Rather, follow it and wait for it to summon a Little Sister out of the wall, then attack it using whatever strategy suits you. If you attack it when it's alone you will not get to harvest or save any Little Sister as a reward for killing it.