Desktop notifications should stay in Notification Center till manually cleared.
I'm using Opera 46 on macOS Sierra. I believe the default behaviour of desktop notifications should be to remain in Notification Center until manually cleared. This is how Safari behaves and other browsers should do the same. In Opera, notifications appear for a few seconds and disappear on their own without any interaction. This makes it impossible to read, say, email notifications that appeared while the screen was locked, since they're gone when the user returns and unlocks the screen.
Firefox has also filed this behaviour as a bug and is fixed in the upcoming Firefox 52.