Did you add the new user to the dialout group?
I think (I haven't done it in a while) you can just add them with:
usermod -G dialout username
and that will add the user to the dialout group. Log out and when you log back in as that user you should be able to use the modem.
You can also use this command to add all those groups you will want your new user to be part of, since I think adduser doesn't add any groups.