We have ours through AT&T U-Verse.
We have never had a single problem, nor do we have to pay for voicemail.
I can log in and look at call logs - incoming, outgoing, missed, and dialed (but not connected). ALL of 'em. Both Onyxx and Jett have gotten busted making calls when they were grounded.
Voicemail is on their server, the same place I get call logs I can listen to voicemails. Honestly, I've got about 10 out there... LOL! We rarely do. Or you can get 'em through the phone, too.
I can also play with my DVR from there, too...