In the current REST API (3.7.0) it is not possible to view the annotated tag message. It is quite useful, because a lot of people save the release notes in this message.
In our company we have a tool, which shows release info based on tags. Because the Stash REST API cannot report the tag messages, we have to check out the repo and parse the output of git -n.

Maybe you can add the message text to the tags method (/rest/api/1.0/projects/{projectKey}/repos/{repositorySlug}/tags)