Instead of making the "Best Friends" list public, Snapchat created a symbolic emoji system.
For example, if you've been snapping a friend quite often for consecutive days, a fire emoji will appear next to your username. My personal favorite is the smirk emoji, which signifies that you're their best friend but they're not yours. I like that the face gives off a smug, mischevous vibe.
I guess Snapchat CEO Evan Spiegel wasn't bluffing when he said he'd find a way to bring back the infamous "Best Friends" list while maintaining some level of privacy.