Some very dense, cold, salty water does spill out. Where the overflows abut the remnants of Antarctic-source bottom waters in the North Atlantic these mix to form the lower North Atlantic Deep Water. The present vigor of the thermohaline circulation seems to owe much to the North Atlantic. Note the near-bottom tongues of cold water extending into the Atlantic and Pacific from the Antarctic. Although the Antarctic upper layer waters are as a whole relatively fresh, some are saltier than others and when they get very cold they can become quite dense.
World Ocean is filled from the densest available Antarctic waters.