Actually there is a teleconference service, from the same company that DC's calls are using now, which offers free teleconferences that you can connect to through Skype -
http://www.freeconferencing.com/ . You can call in with a regular phone or with Skype and no one sees each others Skype name. When connecting through Skype you just call "freeconferencing" and then enter the conference number with the keypad as you would if you were calling from a regular phone.
The only one who can potentially see Skype names and caller-id's is the one who logs in to the online account for that conference number, but it's not necessary for anyone to actually login in order to conduct the call. The leader of the call could avoid that logging in if you want to maintain anonymity. There are some really useful features if you login to the account, like muting specific callers if their line has too much background noise, but logging in is not required for the call to be conducted.