Say, one has a regular preamplifier with a volume control knob, and then one adds a DAC to the stereo system, which has its own volume control knob.
 WHICH VOLUME CONTROL KNOB actually DETERMINES the VOLUME LEVEL- the one on the preamplifier or the one on the DAC?! Or do they just battle it out between themselves,eh?!
Both, in the case you stated.
When you have a preamp for volume control,  the dac volume is set to fixed output instead of variable output.

If you wish to connect the dac direct to amp without a preamp in between, the dac should be switched to variable output for volume control.  

It is usually better to just have one component control the volume rather than two. The dac manual would have instructions to switch between fixed and variable. 
To expand on jon2020’s reply,most stand alone  PA/DACs today have both Fixed & Variable analogue outputs or software based volume control.In a preamp with built in DAC the output from DAC to amp would have to feed through the preamps volume control & the DAC would have no variable output.