Regarding the "hot pot" as shown here – For me, the term that is associated with this style of eating/cooking-at-the-table is "steamboat". Like shown in these images (certainly the ones at the upper/top part of the entire image set), at least when I was growing up and in the milieu I was brought up in, which was more Cantonese/Southern Chinese Diaspora. One talked about going out for "steamboat", not "hot pot"; the expected cooking liquid would be a starting broth or even water, and the cooking vessel would be these center-chimneyed pots that were brought out to the table. The term "hot pot" for us tended to conjure up other versions where a pot is used without the central chimney (e.g. Sichuanese, Mongolian, etc) with or without a divided pot; and/or Japanese-style a la sukiyaki or shabu-shabu and the like.**
We used our own chopsticks (no "intermediate serving pairs") for both the cooking as well as the eating, plus little sieves as shown in some of the images in the Google set above. Nobody got paranoid about "someone else's utensils in my food" 
(besides, the hot broth took care of things...) unless they had done something really objectionable such as constantly licking their chopsticks or using it to pick their nose (not that anyone ever did that last thing). 
**ETA: Or, if still with Cantonese/Southern Chinese cuisines, it suggested stuff like rice done in those claypots - like these...
ETA2: "Fondue", for me and my folks and friends, ALWAYS meant the European/Swiss/German thing , particularly where cheese was used; and those long-handled small two-pronged forks were de rigeur.
