Currently, Bamboo automatically sends build status notification for all builds from Stash / Bitbucket Server type repositories.

Bamboo should provide an option to disable sending build status notification at both the plan / job or repository level.

Workaround

Convert the repository from Bitbucket Server type to standard Git. You will lose the benefits of Bitbucket Server integration, such as push based triggering and branch detection as well as pull request integration and automated SSH authentication management but build statuses will not be sent to Bitbucket for this repository if it's standard Git.