Unfortunately no, the webhook you configure on GitHub doesn’t allow any communication back, from bitrise.io to GitHub, it only registers events to be pushed from GitHub to bitrise.io.
In order for bitrise.io to be able to push back build statuses to your git hosting service (GitHub, Bitbucket, GitLab, …) bitrise.io requires authentication.
To allow/authenticate bitrise.io to push back build status infos (or perform other operations, like auto-registering an SSH key or Webhook) you have to specify a Service credential User on the Team tab of the app on bitrise.io
There’s also a “test the connection” button if you specified a user, which can help you debugging connection issues. Once you select a user in the dropdown please click this “Test …” button to make sure the selected user’s connection can be used for sending back the build status for the PR/commit.
When bitrise.io communicates with a third party service’s API (e.g. to push back the build status) it’ll use the specified Service credential User’s related connections. E.g. to push back build status to GitHub it’ll try to use the Service credential User’s connected GitHub account to do that.
This means that the specified Service credential User have to have a connected GitHub/Bitbucket/… account on bitrise.io. You connect an account to your bitrise.io account on your Account Settings page, on the left side, by enabling the related service’s toggle:
Once the account is connected bitrise.io can start to use that “connection” to communicate with the service.