Private clouds are not having the impact many people thought they would, though private clouds are a valid architecture and have value in some enterprise problem domains. However, public clouds are far more common in enterprise deployments.
You see more hybrid clouds than private clouds, but not the hybrids that vendors are selling. Vendors often paint pictures of nicely organized architectures that use hybrid cloud models pairing private and public clouds, but they are very tough to deploy in the real world.
Instead, IT shops are creating their own hybrid clouds to solve their problems. These hybrid clouds are the result of an organic movement driven by the rank-and-file enterprise IT people who simply want to solve their issues using the best technology and approach.
Let's call this approach the pragmatic hybrid cloud.
What this movement has discovered is that you can combine the public cloud and modernize some of your legacy systems to be more cloudlike. You don't have to start from scratch with a hybrid private-cloud/public-cloud architecture, converting your legacy systems to private clouds as the cloud vendors would prefer.
In other words, IT is adapting its existing datacenter technology where it can to be more cloudlike, and IT is adopting the public cloud where it can -- and tying the two efforts together where it makes sense.
What does this approach look like? It could be a legacy Oracle database running on a mainframe that interacts with processes and data sets that are now run on public clouds. Or it could be a traditional client/server application running on the enterprise network that now redirects to a massive big data system running on a public cloud.
This ad hoc, pragmatic approach is very sensible. It makes the most of what you have, reducing the need for new resources and letting you transition to the cloud at a pace you can handle, both in terms of cost and time.