If you go to Settings > Portal Settings > Setup

If you would like to hide the names of players on certain reservation/lesson types, just use this box here:

For events, you can choose to SHOW REGISTRANTS on each event that you create:

Create Event > General Setup:

To hide player information for widgets, it's the same idea. You'll go to Settings > Widgets

You can choose to Create a Widget or simply EDIT a current widget

Just scroll to the bottom and CHECK or UNCHECK the box to either show or hide the player details (reservation type, names, etc.)