Fields in Jira are either required or not. For the entire life-cycle of an issue. (This is a known weakness in Jira, but it's not obvious at first). You control that with field configurations.
Screens are pretty much irrelevant here, all they do is say "show these fields to users if they're on this screen".
The work-around is to use "validators" on the workflow, so that a user trying to leave your field blank when going between status is thrown back to the screen with a "this field is mandatory" type message. There are no validators like this built into Jira, but many highly recommended plugins all provide them (Jira suite utilities, jira misc workflow extensions, etc)
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