Since 1974 fiendish creatures have evolved, gaining and losing many powers and spells down through the ages. Through all of this, the mechanics of possession are ellusive.
Please note all editions of D&D have had careful descriptions on possession when needed - note the (in)famous ghost. It explains how the living target-creature would lose control, which stats change (or not) and how the etherial form phases into a material body. The mechanics are exacting and have really effective game-play usage. It is great.
Contrast this with the 5e Shadow demon: in the writeup they lay claim to some ability to possess targets - yet mechanics for this ability seem to be missing. Most other fiends seem to be devoid of all mind control powers all together. Even the almighty Pit Fiend's only mind alteration does not extend beyond causing enemies to be a bit scared... for a possible six seconds. On a failed save. Once a day. Provided one is not otherwise immune.
It would add a lot to a campaign to have anyone, characters, player characters or even the players themselves (open for anything here), possessed by a demon. Note that StackExchange even features a brilliant write-up on how to be rid of a demon once one has been taken over - this is a Step-By-Step / 'How-To'. It is very good.
Long story short: how do demons possess (player or non-) characters? Where are the mechanics listed?