Simplest way - twitter allows "protected profiles", where only users you are following can see your updates.. This is basically a whitelist of people who can see your statuses..
If you wish to allow multiple users to post from the same account, without hardcoding the twitter account into the application.. you could create your own API, essentially just a proxy for the twitter API..
You could then add your own level of authentication over this, so each user would have their own account (and you don't give out the shared account's login details)
In pseudo code, the application would be something like..
if request['username'] not in ['bob', 'alice']:
raise AuthError
if request['password'] != ['theuserspassword']:
raise AuthError
twitter_api = TwitterLibrary.login("sharedaccount", "secretpassword")
switch request['api_method']:
case "getPublicTimeline":
return twitter_api.getPublicTimeline()
case "postStatus":
return twitter_api.postStatus(request['something'])
Final option I can think of - you could run your own Twitter-like site.. There are plenty of "twitter clones", such as status.net (which is the code that runs identi.ca)
status.net and several other similar projects have Twitter-compatible API's, so you could quite easily take an open-source client (NatsuLiphone for example), and, with permission, rebrand and modify it to use the URL of your own site.