If I have two different PSN accounts on the same PlayStation, are people from account A able to see the games downloaded or bought on account B, and vise-versa? For example, if I buy Destiny on one account, can people see that I have purchased and played it while they are on the other account? If they be able to play Destiny from the other account or will they have to be on the account the game was bought with?
All accounts can see which games are installed on the PS4 through the main menu dashboard and the library screen. If account A buys a game, other accounts on the console can only play said game, if account A has activated the console as their primary PS4, which can be done through Settings -> Account Management -> Activate as Primary PS4.
This means that all games account A buys and some benefits of its PS Plus subscription (like online play) will be shared to all other accounts on the console. If the PS4 is not set as the primary console, other accounts can still see installed games, but they can't start them. Even though games are shared, each account will have their own save files that are not transferable to other accounts.
Other accounts can not directly see which games account A has bought or played as long as they're not installed on the console, but they could see through the "What's New" tab, where games played and reached milestones are shared to friends and following accounts, or by the account's trophies if sharing these things isn't turned off in privacy settings.