Do the destination tickets in Ticket To Ride require that the player completes the routes, or only that the tickets are completed?
The holder of the destination card has to connect the two cities with their own trains. The route can be as roundabout as you like though, as long as they are connected.

In the Ticket to Ride: Europe version, you can build stations which then let you count another player's route as if it was your own for the purposes of scoring destination cards.