If you only have 1 profile on your Switch, all additional local co-op players will be using copies of Player 1’s profile to play (guest profiles), which means the progress they make unfortunately can’t be saved. Only the progress of Player 1 will be saved.
But if you create additional local profiles on your Switch, you will be able to save the progress of every local player! After creating additional local profiles, all you need to do is go to the Character Select screen after creating a lobby. Then have another user join and have them press the "Y" button before readying up. This should allow the other user to select a different profile to use instead of the guest profile and all progress should now be saved for everybody!