Public sharing is automatic:

m4Some projects chose to make badges earned in the system public by default. This has different consequences for motivation than when students can choose when and how to share their collections.

For more ways of how badge systems have chosen to display badges, please see the following page: learners can choose to display.

Mozilla Open Badges
Digital Badges are web-enabled credentials of learning or accomplishment. -Erin Knight, director of the Badge Alliance
Badges contain detailed claims about learning, links to actual evidence of learning, and they're shareable over the web. -Dan Hickey, DPD Project Lead Investigator
To me, digital badges represent the bridge between formal learning & informal structures. -Alex Halavais, DML research competition winner
Open Badges can help people tell a verifiable story about their accomplishments. -Nate Otto, DPD Project coordinator
Regardless of where you start, it’s more than likely you’ll end up somewhere other than your intended destination. That’s okay. Systems are living things, and your badge system needs to be flexible. You must embrace a bit of chaos in its design. -Carla Casilli, Director of Design + Practice at the Badge Alliance