Hmm, not sure. This ended up in the Note 4 page, which I think will limit your audience severely. I doubt your issue is limited to Twitter on the Note 4 specifically, though.
Perhaps try asking here?
https://forums.androidcentral.com/android-apps/ - it's a more generic app support thread and you might have a larger group of people noticing.
OMIGOSH, I FOUND IT!!! This was driving me bonkers on my Samsung S7. OK, so, go to your phone settings. Select "Apps". Scroll down to Twitter, select "Notifications" (under App settings section). Hit the little "i" Info icon to the right of the title "Twitter". Then select "Notifications", scroll down to "Push Notifications", scroll down to Ringtone, tap to select where to pull your ringtone file from, and pick your ringtone!
I realize that was a very roundabout way of getting there. But it was more intuitive initially, as I was poking around in my settings and I overlooked it in the app...You can also open the Twitter app, hit your profile pic, select "Settings and Privacy", "Notifications", scroll down to "Push Notifications", and follow the remaining steps above.