For a short visit like a social call, your driver (and possibly footmen) would stay with the horses outside. For a longer event, like a dinner party, ball or night at the theatre your carriage would drop you off, then return and collect you at a pre-arranged time, which is why grand formal invitations still say 'carriages at x time' so you know when to tell your coach to pick you up.

The houses of the kind of people who had their own carriages had stable buildings attached, usually around the back and accessed from a side street. These buildings included stables for the horses, a place for the carriage and accommodation for the driver, groom etc on an upper floor. Lots of these private urban stables still exist, they have mostly been converted into homes and are known as 'mews houses'.