Do journals allow authors to bring their own reviewers Some papers are published under category "communicated by ...". Do this mean the author contact this person to review the paper instead of editorial board? My question is: do journals generally allow authors to bring their own reviewers for evaluating the papers? It seems that some journals allow that.
There are some journals that ask authors to suggest reviewers. There is no expectation that these people will necessarily be asked, but it gives the editor a backup at least. There are some journals that ask authors to suggest people who should not be reviewers, though I think that is a bit less common. 

Most journals, however, have their own pool of reviewers and the author has no input at all into who will review the paper. Most of those use blind reviewing in which you never learn who reviewed your paper. 

But in the case you mention, it is likely that someone other than the author sent the paper to the editor. That may actually be required in some situations. There is no expectation that review is then suspended.