Restaurants in the US don't bring a machine to the table, which I understand to be commonplace elsewhere. (At least, I have never seen one in my entire life.) The waiter delivers a paper bill to the table, often in a billfold. The diners get together cash/cards (and very rarely, checks) and put it on top of the check or in the billfold.
If you're splitting the check, you have to tell the waiter who to charge for what. If you're lucky, everyone ordered their own meals and drinks and nothing was split. Then you can usually just say "this card is mine, this card is hers, this card is his" and it works. Otherwise, you usually have to break out a pen and tally up everyone's share and write it on the back of the bill or somewhere like that.
Then the waiter goes away with your cards to the register (or a computerized POS system), runs the charges, and hopefully doesn't copy your card for their own fraudulent use. They get separate printed receipts for each paying diner and bring them back to the table along with the cards and any change (if people paid in cash).
People paying in card add their tips to the receipt and sign it, and people paying in cash leave their tips on the table. Then you're done.
Suffice it to say that this is a huge pain in the ass for everyone involved, and takes time that the waitstaff could be using to provide better service to other customers. Also, especially for smaller restaurants running on an extremely narrow margin, the credit card fees for splitting a table too many ways can be financially unacceptable.