I want to still be able to use iMessages on my MacBook but I want to stop my messages from displaying on my screen (for privacy reasons). It seems you have to completely sign out from all of your iMessage accounts and turn off iMessage to make this happen, but is there a way to keep iMessages functional and just change the display options?
Have you tried the following from Apple's website:
OS X Yosemite: Turn off notifications
If you don’t want to see notifications in the top-right corner of your screen, you can turn them off. You can turn off notifications for certain apps, or turn off all notifications during specified times by setting Do Not Disturb. For example, you can turn off notifications sent from the Calendar app, or turn off all notifications for when you’re asleep.
Turn off notifications for an app
Choose Apple menu > System Preferences, then click Notifications.
Select the app on the left, then choose None below “[App] alert style” on the right.
The app’s notifications won’t appear on your screen. If you also don’t want to see the app’s notifications listed in Notification Center, drag the app down, to the Not in Notification Center list on the left.