How to judge an amplifier's performance with a "powered-woofer" speaker?

I think that the OP brings up a great point in that you dont necessarily get as good an indication of the true amplifier total response with a speaker with a powered woofer. A good scenario would be if you were using an impedance sensitive amplifier like an OTL. 

You can't have it both ways, obviously. My point is that the typical audiophile would think that not much would happen to the bass response of a speaker with onboard amplification for the bass. That is wrong, and typically the perception of the entire speaker's performance, including the LF, is reoriented through changing amps. 

To address the concern of the OP, if, perhaps he is considering the amp in question, I have not found typically that an amp that performed better with a fully passive speaker did more poorly with one having powered bass. In my experience, usually, an amp is superior/inferior in entirety. It is quite uncommon for an amp that would not drive a passive speaker well in the low frequencies to be superb in the mid/upper. So much of the performance is shaped by frequencies above 100Hz that you will by and large have the character of the amp regardless of the speaker being active/passive bass  (Some will disagree, no doubt, but I am not interested in arguing that point). Imo, one might think that a lower powered amp would be superior in such an application (powering the M/T) as a speaker with powered bass. No, you have no clue until you compare, and when I have done such comparisons, the superior amp with passive speakers has been superior with speakers having active bass. I have typically enjoyed the headroom afforded by higher powered amps to lower powered in such applications.

So, perhaps that summary helps the OP understand why a reviewer would pursue such a system and comment on the bass. Technically it is correct that the internal amp would power the bass, but the experience is holistically different, and valid when comparing amps. 

The same holds true of powered subwoofers; the perception of their performance is changed strikingly by swapping out the mains amp, even if no power cord or signal cable is changed to the sub. Again, amps that did better with solo main speakers typically also did better when the subs were added. 

That is not to say you can't massage the sound of the passive or active to make one sound better than the other. 

Hi guys,
just want to thank you all for the responses. There's a lot to digest here.

Turn a powered woofer into a positive by reducing the demand on and for an amplifier.  
For example you can power them with a smaller amplifier than with a typical passive woofer speaker.  
For example a tube integrated Primaluna amp driving a powered Goldenear Triton 1 powered tower speaker sounded excellent- tube soundstage and midrange bloom with the dynamics and bass impact of a robust solid state amp.  
You can get large room sound with a modestly powered decent quality amplifier.  No more 80 lb. behemouth amps required.  

Coming back to the OP's question:  Amplifiers are not independent commodities. You don't gauge their worth and put them in your closet, waiting for the to appreciate for years before you take them out and sell them still in the box.

Who cares if an amp performs well on speakers you don't own?

Amps are part of systems, tightly coupled to the speakers.  When evaluating the performance it is the system behavior that matters, not a test bench and not in a system you don't own.

In general, yes, most amps whether solid state or tube are more impedance dependent than we'd like to believe.  Using a powered bass section in a speaker will stress this particular issue less than a traditional multi-way speaker would.  It does not by itself change the value of an amp, unless you are a reviewer, of course. :)