Short answer is No, you don’t have to use a public domain name in order to be able to preview or to even share your own Clickfunnels funnel.
I’ve seen some of the biggest experts using Clickfunnels that are still promoting their stuff (products, services…etc) using clickfunnels own sub domain…
Technically NOPE, you don’t need to have a domain name (like mydomain.com) in order to publish or even promote your Clickfunnels’ funnel or landing page, since Clickfunnels automatically assigns you a free subdomain that you can use directly, without the need to any extra configuration.
But taking it from a performance, psychology perspective, People tend to trust domain names (especially short ones) compared to the long domains or free-subdomains.
And if you are one of those focusing on SEO to drive traffic, then a root domain is a MUST.
You can find more insightful information about how a domain name can affect your business on this wiki article by @yassine: