The Apple Watch is a somewhat private device by its very nature, since it sits on your wrist, so having Messages and other notifications appear with full detail is probably not going to be a problem for most users, particularly since notifications are only received when you’re actually wearing the device. If you’re looking for a bit more privacy, however, Apple does provide the option of suppressing notification details until you actually tap on the Apple Watch screen.
To do this, simply go into the Apple Watch app on your iPhone, select Notifications from the main menu screen, and toggle “Notification Privacy” ON. With this setting enabled, you’ll still get notifications for things like Messages and emails on your wrist, but the text and other details won’t be shown until you actually tap on the screen, shielding them from curious onlookers.