My first thoughts would be to place a prominent notice telling your visitors about the new changes - splitting the site in two - with prominent links inviting your visitors to select one of the new sites. Similar to having a visitor select a particular language version, except you're having them select which of the new sites they wish to explore further.

Assuming no conflict, aside from the notices I'd leave the existing site intact to begin with... display the notices site-wide via CSS if applicable, and ask visitors to update their bookmarks/favorites using one or both of the new sites.

Gradually MOVE content from your existing site to one of the new ones (avoiding the duplicate content issue). Simultaneously, replace the content on the old pages with 'time delayed redirects' or give them a choice option as I already described - explaining the move/changes (you don't want the old pages to suddenly just disappear altogether, that would cause a rash of dead links and lost search engine traffic).

If you have a user base you can email, then by all means, let them know via email.

When traffic to the new sites has picked up and the old site has dwindled to nothing, just show the "selection notice" on your old home page, linking to the new sites for a while... and eventually just drop the old site.