Amazon will go from using many competing companies to using their own shipping service… maybe you have a different way of looking at it, but that's less competition in my book.
The locals will still exist (but many of them aren't good, as parent said. As both a shipper and a recipient, I've had bad experiences with OnTrac for example), but this will create another national/international carrier, which I think is a plus for competition.
Amazon can provide a competing shipping service. If they give me a choice, I'll pick the one that best fits my needs. It might be theirs; it might not be. If they don't offer that choice, and force me to use theirs, and theirs fails to deliver according to my criteria too often, I'll shop elsewhere. How did that limit competition?