You're playing each crossword at the same time as everyone else, racing to complete it first. You can't see what you've got right, until you correctly complete the entire grid. But you get some fun feedback on what other players are doing: a cell turns green if one other player has correctly solved it, orange if two other players have, and red if three or more other players have entered the correct letter in that cell.
Chat is emoji-only for the first half of the game (i.e. 30 minutes for the front-page, 5 minutes for the mini). After that, it unlocks and you can chat freely.
If you're not done when the next crossword is generated, you can just stay on the current page for as long as you like and keep working to solve it.