the pod HD "full" models have poweramp distortion. the "preamp" models do not (they just model the pre-amp section of the given amp). "drive" controls the amount of pre-amp distortion, which is usually what people use for their main distortion stage.
if you double click "enter" with the cursor selecting the amp model, you should see the normal bass/mid/treble/etc. controls. if you press the right button a few times, you'll get the "deep editing parameters" (or DEP's) including "master volume", bias, sag, bias x, and hum. "master volume" controls the amount of power amp distortion you get for the particular amp model.
cranking the volume knob on the front of the unit does not affect power amp distortion at all. If turning it up gets you distortion you are distorting something downstream of the amp in the signal chain, possibly an effect in the pod hd, or possibly external gear.
Some amp models' power sections will compress/distort more than others. So the effect will vary per amp model. Cranking it up on some amps it will simply compress the tone, while it may turn others to mush. Some might get a nice smooth distortion, while others may be splatty and dirty.
Also keep in mind that the bass/mid/treble/presence controls are part of the pre-amp and will take effect before any power amp distortion. If you are getting a fair amount of power amp distortion from the amp model, these controls will have more effect on the nature of the power amp distortion rather than the final EQ of the tone. So cranking the bass and leaving the other knobs at 12:00 may result in a fuzzy/muddy distortion, instead of a bass heavy tone.