Status updates do not automatically cross-post, but duplicate postings can occur.
Cross-posting is when the same status update is published to multiple social networks. This occurs when more than one social network has been selected from your profile picker to publish your status update.
Duplicate Tweets occur when you post to Twitter and Facebook using Hootsuite and your Facebook account is set to automatically post to Twitter. Unlinking Twitter from Facebook will prevent duplicate Tweets when using Hootsuite.