Well I suppose the first thing to clearly establish is that registering users is not a function of SP. This happens within the WordPress core of your website. All Simple:Press does is work with those users and react to the data we are sent.
You said at the top that you would have public forums and member only forums. This is, in fact, exactly how our website here is set up. To do this then you will need to be using some sort of membership plugin. Decent membership plugins can assign users to different WP roles and you can map Simple:Press User Groups to different roles (see the user group admin for SP – Map Users to User Groups. So it starts with the membership plugin.
The FAQ below might help you understand more. This one was written around the WishList plugin but most membership plugins are similar. The main thing to look out for is that the membership plugin does obey WP standards and fire off the standard WP action and filter hooks. The authors should be able to conform this ort tell us which you are using and we may well have encountered it in the past.
https://simple-press.com/documentation/codex/faq/how-to/how-to-configure-simple-press-with-membership-plugin/