Faced to the need of saving access to issue for Reporter and Watchers when issue moved between projects. Reporter and Watchers don't have access to the new project.
How to solve such case automatically in jira server version?
If you want them to see the issue after the move, you have to give them access to the new project.
How manage such result: wathers and reporters have access to new project but only to those issues?
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You can't with watchers, the point of them is they are external to the project but have at least read-only abilities to see it.
For Reporter, add "reporter" to the browse issue permission. Two caveats
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