If only a few people need to share with external users, then putting things in Office 365 is a cheap option. Store things that need to be shared on O365, and everything else can stay on prem.
Another option that is technically feasible, though IT will say no, is to create domain accounts for those users, and allow them to access the site the same as other users. Clearly, this has some pretty significant security implications, but sometimes "external users" are long-term, trusted individuals whom IT doesn't mind giving an account.
edit: if you want to go the official route of setting up an extranet site without giving users AD accounts, an excellent resource is the SharePoint FBA Pack, located here. Be sure to see the "important" section, as it has a link to an excellent guide for configuring SharePoint for Forms authentication. This is not a simple thing to set up, and there are quite a few challenges for using it. As a result, my current preference is the separate O365 site, but again, the FBA authentication works as well for an on-prem solution.