While the dropdown looks more simple, I would go for the second option. The reasons are:
- the use of a visually and functionally separated module should help users to focus on the task of selecting the names from the list (you can use a semi-transparent layer over the rest of the interface while the shuttle control dialogue is open)
- the list of already selected users to the right will not shift the dropdown list down after adding a person
- the lists height can be way higher in this approach, as it is triggered as a modal view, while when using a dropdown it would depend on other elements of the interface (dropdown would have to adjust to the rest of the interface). The higher the lists, the less need of scrolling inside of them and more effective selection review made by the user.
- actually, you can use almost full application window size to display this modal if necessary
- aditional buttons [Select all] and [Remove all] can be very handy
I can see just one disadvantege, which is triggering a new view in general, but I think it is not important considering the pros.
By the way, in my opinion it will be more natural if you put the Cancel and [Done] (in this order) in the bottom right corner. Refer to this question: Submit Buttons on the Left or the Right?