The main complaint I've read is that it is uncomfortable. In fact the salesman from the US website hardly wanted to sell it to me hinting that most people find it intolerable and that I should instead first try their smaller nasal "Alitostent" stent that only goes three inches into the nose and increases nasal breathing but does not prop open the soft pallet like the longer Alexostent does. That one does not have any sleep apnea study demonstrating its efficacy though so I didn't want to bother. I persisted and they kindly accepted my $800 which got me what looks like a fancy Chinese finger trap that I then set about trying to insert into my nose and down into my upper airway. It was really uncomfortable and as I gagged and mucous poured out of my mouth and nose and quickly said forget this and didn't touch it for several months. I eventually tried again and found inserting it in just two thirds of the way was much more tolerable and after sleeping with it for a couple of nights like that I got used to it and i could put it all the way in. I noticed some blood on it for the first few days too so it does scratch up your airway a bit. Maybe that is why it's so difficult at first. I don't mind it now though I would rate it as less comfortable than CPAP and MAD. Depending how far in you put it swallowing can be a little uncomfortable. Wish I could find a more elegant way to say that.
I can't really speak to its efficacy since I didn't do a sleep study with it. Note even their cherry picked results were not that impressive. CPAP and MAD both work better. But keep in mind you can combine all of these therapies so if you're not getting the results you want and have the time, money and patience there is no reason not to use more than one. I use it with a CPAP machine, but as the end of it hangs out of your nose a bit (they have you tape the end to your face) it messes with the mask seal just a bit. I can say using it with a CPAP works better than just using a CPAP based on the machine apnea detection. And interestingly it completely stopped my snoring. This really only helps out the upper airway. If your problem is a big tongue I doubt it would help much. I use it most nights. Sometimes I'm too tired and just don't want to bother. It only takes about 30 seconds to insert now but if you're tried it's not always worth it to me for the small improvement. It's definitely expensive for what you get and insurance will not cover it in the US as far as I know.
You can read about it and see the small study they did on it here. http://www.alaxo.com/alaxostent_eng.html