I have Stax Omega II with the tube interface/amp. I have not heard the phones with other amps (I ordered, and I am waiting for the Blue Hawaii Special Edition amp). My comments are not about electrostatics in general, but about this specific model. Like any transducer, it has a characteristic sound (coloration). In other words, regardless of the music and source, there is a "sound" that is pervasive. To me, it is an upper midrange prominence that I would characterize as the sound of plastic. To some extent, I hear that with panel speakers, but, with speakers it is usually lower in frequency. Every time I start a listening session with the phones, I notice the coloration, but, in a very short time I get acclimated and it is not that big of an issue (I also use horn-based speakers which have their own coloration issues).
The Stax phones seem to not have as much dynamic punch as other phones, but, that may be just an impression that comes from their lack of overhang and bass resonance. Speakers and phones that resonate (as well as rooms that reverberate) can give the impression of more bass punch.
The Stax phones are extremely revealing about characteristics of upstream components, because of their extreme clarity, and this, I suppose, can be a negative in some situations. I didn't like the dry and slightly brittle sound I got with a Levinson No. 32 preamp, and while I get much better results with my Emotive Audio Epifania, I would prefer a sound that is a bit warmer (more upper bass, lower midrage), but then, the sound with my speakers wouldn't be optimized. This is, a very demanding component.
I've had my phones for some time, and so far, I've experienced one operational problem. The volume potentiometer for one channel has gone bad at the high volume end of its range so I cannot play the phones that loud. Even with 13 db of gain from my linestage, I cannot get sufficient volume when playing vinyl. I could bypass the volume control, but, I keep thinking my Blue Hawaii amp will be shipped any day soon.