@DrSbaitso
I don't know what I usually do. I think I let the waiter/waitress lead, so they can do whatever easiest for taking it down.
It does seem to make sense not to keep going round the table, though.
Especially if you're eating out with my mother, who is of her generation and thinks that it's always rude to say what you want in terms of food and drink. (For instance, if someone offers you a cup of tea at their house, it must be declined at least three times with rising protestations of sincerity before you accept. I think she nearly died of shame when I once said to my aunt, 'Yes, I'd love a cup of tea, thanks'.)
In her view, this is also the case in restaurants, even when the person asking what you want to eat and drink is a waiter whose job is to ask what you want, so she does this weird pantomime of pretending she can't decide because it all looks so delicious, and hums and haws, all smiley and deferential, as if she'd prefer to waiter to choose for her, because that would be more ladylike. Meanwhile, some poor server is just trying to take the order.
And then it starts all over again with the dessert menu. Waiter offers the dessert menus. My mother invariably protests. Oh, no! She couldn't possibly, she's eaten so much! She'll have to go on a diet in the morning! We'll all have to go for a long walk afterwards to 'work it off!' Meanwhile, everyone else has read the dessert menu and said 'The panna cotta, please.' Whereupon my mother looks shocked at everyone else's greed and rudeness, and then order the chocolate tart as though forced into it.
If individual courses were ordered, it would be midnight before my mother finally settled on the chicken. She always settles on the chicken.
This, added to the fact that my father, despite having eaten out with us for years, still doesn't seem to grasp that you ask for the bill and they'll bring it to your table after you've finished, and as he's a gobbler, has been known to leap up when he's finished his main course and everyone else has taken two bites, and try to pay a waiter walking past with plates up each arm, because he thinks the other four people at the table are dawdling. 
Oh, that felt good to let it out. I love my parents, but they are not soothing dining companions.