It's been 2 years. They have no specific plans to release the server code.[1] They won't say what's been done or has to be done.
[1] https://github.com/Pocket/extension-save-to-pocket/issues/75
[1] https://github.com/Pocket/extension-save-to-pocket/issues/75
No one ever talked about open sourcing the service that the pocket client relies on, as far as I can remember.
Mozilla's claimed principles include privacy, interoperability, and free and open source software.[1]
2. Is there an open source implementation of compatible server software, that you could use to run your own server?
The answers to these questions will tell you whether Pocket is open source.