Let's put the answer in a different way:
SME does not support this function. Users are added, changed and deleted by administrators. Users have limited ability to change their own accounts, but certainly not add or delete them.
That said, this sort of functionality is certainly workable, but it would have to be done using custom created code. You would have to write a PHP or other such application to make all of that happen.
This does bring about the questions "What?!", "Why?!" and "Are you out of your mind?!" but in the end, it's your machine(s). You will have to deal with the hordes of spammers and hackers and crackers and criminals and the like when you let an anonymous public create accounts and send email through it. Presuming you are not a programmer yourself, I recommend befriending a programmer somewhere and talk it over with him.