I get my voicemails sent to me via email. It is an option on my cell provider. It is handy for me. Easy enough when I am at my desk to just get them there, click and listen. Of course, I'm not doing anything nefarious.
Maybe that is what is going on with the DNC. Maybe it was not a hacked voicemail system, but simply listening to the attachment to an email, no different than opening an Excel attachment or something like that.