There should be a box above responses within a need to type in a name and click add. The same as you would add the need to a user group, we should be able to add users to needs without assuming their identity and then having to sign out and back in. When there are larger groups, this becomes an impossibly long endeavor.
You can add users to opportunities by going to the Volunteerism > Responses area of your site manager panel. You can find instructions on how to do that here.
Would this solve the problem, or do you think a box on the opportunity page is still needed?