Are reviewers allowed to discuss their review with each other? Suppose you find out that someone you know well or work with is reviewing the same journal paper you are reviewing (i.e., finding out this information on your own, and not via the journal or editor, assuming a blind process). 

Can you discuss the paper? Can you discuss your reviews?
If it's a blind process, you're not meant to find out. As various people have pointed out in the comments already, discussing your review with them is therefore a big no.

One of the main principles of reviewing by several people is that you get independent opinions, so that if a particular (maybe well-known and well-respected) person doesn't like the research, they can't just make it disappear by convincing everybody else that it's bad.

For some conferences (and maybe journals?), you are able to discuss your review with the other reviewers after the initial submission. But in this case the facilities for doing this are provided by the submission system. There are two important differences to the situtation you've described though. First, your initial review will not be influenced by the discussion and second, there is a record of the conversations and changes made.