With version 32 of Google Duo, Cody discovered an activity that would let him export the call log history to any app on his device. Despite the feature being functional, it wasn't visible and had to be triggered from ADB through your computer. Now, the option is getting its own UI element, although it's hidden where you don't expect it to be.
Open the Google Duo app, tap the overflow menu on the top right, and choose Help & feedback. Tap again the overflow menu there and you'll see the Export call history option at the bottom. If you choose it, you'll get a Share interface so you can either copy the history to the clipboard, email it to yourself, save it to Drive, or create a note with it in Google Keep, for example. The format is still the same one that Cody described: a CSV list with the date and time of the call, phone number, direction, and duration.
The option appears to be live in versions of Google Duo going as far back as v32, so it seems as though the feature was enabled remotely via a server-side switch.