Bitbucket is ok for private repos. I think private repos are relatively secure there because Atlassian has much of reputation to loose if there is a security breach. Another plus for Bitbucket is the integration with Jira and other Atlassians tools.
Beside that plus points I will rather go to GitLab.com (more features, better UI and integration with 3rd party) but my trust in private repos is lower there than on Bitbucket. I wished we could measure security somehow for private repos between Bitbucket, GitHub and Gitlab.com