Create Twitter app: @TheStalkrrr or something equivalent
Follow / Unfollow page (enter username to follow, click on existing to unfollow)
User account management; create user groups, admins, add users to your group
Separate credentials for user login so as to not necessarily tie every account to Twitter
Implement Oauth for login, save credentials of all users in a group for proxying their requests together
Timeline display page: show all accounts being followed by @TheStalkrrr for each user
The general idea is to anonymize Twitter follows and group together timelines so that people can read timelines of others without letting on how many people are reading, or who is reading, their tweets. This might be against Twitter terms of service, or it might be a matter of interpretation. This system doesn't have to be used by a single anonymous account, one could choose a representative to be the follower, and then leverage the Twitter credentials of everyone in their Stalkrrr group to make additional REST calls or whatever.