I have the same problem! Never had this problem before... today in 3 games!!
Bug, it's says that it's my move, but it's my opponent's move.

Magic number three! Three broken, the other seem to work. At least those in which my opponents made there moves recently.

I've read in a related thread it might help to ask your opponents to check if they can redo their moves.

I've read in a related thread it might help to ask your opponents to check if they can redo their moves.
I wonder who will lose rating points if the opponent doesn’t redo the moves when time runs out.

I don't have the problem on my phone and there is my opponent to move. Seems that he has not moved yet. I'm confused...

I'm not sure but would assume the staff would be able to reset those games or fix the ratings.

The bug appears to have caused any Daily moves made while it was running to not actually have made it to the back-end and a move notification sent. So any moves made during that time will likely need to be remade.

I don't have the problem on my phone and there is my opponent to move. Seems that he has not moved yet. I'm confused...
By the looks of it, this problem only occurred for players whose opponents made their moves in a certain time frame a few hours ago. No need to be worried if you're not having any issues.

The bug appears to have caused any Daily moves made while it was running to not actually have made it to the back-end and a move notification sent. So any moves made during that time will likely need to be remade.
Ah, thanks for this clarification!

Could this be a start of a chess virus?! I might as well use the app on my phone whilst this is going on.

Could this be a start of a chess virus?! I might as well use the app on my phone whilst this is going on.
Nope, no virus here. They had an update on their servers which turned out to be bad. Now they've set back the servers to the previous version. The weird notifications should go away if the opponent does his move again. At least, that's my understanding of this issue.

This is a serious bug for chess.com to have introduced... this is why one tests updates before publishing them to the production server.
Earlier games did not save moves either.
Double-check all those games where you might think it's your opponent's turn. It may be YOUR TURN and your clock is ticking.

The bug appears to have caused any Daily moves made while it was running to not actually have made it to the back-end and a move notification sent. So any moves made during that time will likely need to be remade.
What a DISASTER for chess.com.
Who's running the place? JDCannon? Someone should have TESTED those updates before publishing them. Now ~24-48 hours from now games will be TIMING OUT all over the Daily chess universe and no one will know why unless they're paying attention.
Why isn't there a SYSTEM-WIDE NOTIFICATION of this issue to ALL PLAYERS?!?!?
Same problem here on the chess.com site. My iOS app seems to show only my moves, the website shows others where it shows it is my opponent's turn but says it is my turn in the list.