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This works on my end. Try using the official keys (remove account from Twidere, enter official keys as linked in your post, re-create account in Twidere). There has been a point where Twitter removed some DM support from the API for third-party apps... with one of these keys, Twidere looks like the official app on API level.
I've just switched from Twidere(*) keys to Twitter for Android official app. ones (as linked in your post). And it works fine: DM, even with pic., sent&received, notif. with answers&fav.
Question: are your connected in the same time with the Twitter for Android official app. (from the Google Play store)?
(*) v. 3.7.3 from F-droid running on Android 6.0.1