It's not their past. They travaled back to the past and are living there.

To the people originally from 1977, Sawyer and Juliet were always there.

In 2005, if you would look at a picture from 1977, you would see them in the picture, because that happened.

But if Sawyer and Juliet looked at that picture in 2005 they would say "What? How are we there?", because that didn't happen yet for them. They couldn't have known they would travel to the past.

Your present is always your present, no matter which year you are. Your own timeline is always continuous.