Should I "push" or not?
Well, yes. Otherwise you still won't be able to access it from the outside!
I'm afraid to let the repository unusable or mess up others work
No risks. A git repository has no "files" per se but is a directed acyclic graph of commits; what you do when you add a branch is just create another branch to the graph; it will not affect any other branch.
The difference between the remote and you is that you have the "contents" of a branch "checked out" in a directory; the bitbucket remote doesn't (it is what is called a "bare" repository).