Noise issues with dsp like minidsp.is almost always gain structure related. As said already, for convenience most people use an analog volume control that comes before the minidsp. So now you take an analog signal that has been significantly attenuated (normal listening levels) but the self noise of the preamp is not attenuated, and do an A/D conversion on that reduced signal. Now you've also effectively reduced bit depth early in the chain and all digital processing is done on that reduced input.
The ideal way (not the convenient way) is to maintain a full scale signal into the minidsp. In digital is even better. Processing is done on that full scale input, and volume attenuation is done after processing just before amplification. That could be analog or using the digital level control built in to the minidsp.
I recently went through this design exercise, and it was pointed out to me with the requisite supporting math that truncating the digital output in the dsp just before the D/A will almost always be as good as or better than analog attenuation afterwards, which was not what I expected. Small amounts of digital attenuation are "free" and usually when any actual bit truncation starts the level is so low you aren't listening critically anymore. Exceptions might be with highly sensitive speakers where the levels will almost always be highly attenuated, in which case I'd suggest using a passive pad as mentioned earlier whether you are doing analog or digital volume.
That being said, digital volume isn't easy when you are using more than one minidsp or other brand. In fact, its downright hard. And, despite what's ideal, for many speakers and many listening needs, upstream analog works well enough to not cause audible issues. That's why tons of people with minidsp and other digital processing downstream (like built into amps) don't complain. High efficiency speakers or out of the norm gain structures might bring the issues into audibility.