While Yammer seems an interesting spin on things, I'd much prefer to have this functionality contained within Twitter in the form of private groups - purely from a convenience point of view if anything.

If the Twitter dev team were able to give me everything I wanted, then I'd want the ability to start a private group with people, where updates into that group are protected from my public timeline.

I'd also like the ability to add custom tabs onto my twitter UI so that I could get instant context of any conversation happening within my group, while maintaining the ability for these messages to get forwarded to my mobile device if I wanted.

I would pay for a service like that, absolutely.