It is difficult to get the balance right, however overall I don't believe the PM should expect to see every communication. It is vital that any decisions, emerging risks or issues, completed actions, or anything that needs escalation are communicated. However, different people will interpret these in different ways so it is important to give guidance and either ask for more or less, depending on how individuals communicate.
Some people like to tell you everything, on the basis that "you can't blame me because I told you what I was planning to do" - but they don't appreciate that their 20 emails per day are a small proportion of the total that you receive. Others will tell you very little as "you are so busy that I didn't want to bother you with trivia", despite their actions or decisions being non-trivial.
I try to draw the line about being copied into deep technical discussions between technical people, and ask to be removed from the distribution list until they have something that fits my communication criteria. Having said that, I would rather get too much information than not enough, as it is a small overhead to scan an email and delete it when compared to having to step in and resolve something that has blown up out of all proportion, and that could have been nipped in the bud 3 days earlier.
So, coming back to the original question, I suggest that there is no single best practice model, and the PM should set out clear criteria for communications - and be prepared to have to coach the team to comply with his or her expectations.