Watchers field is one of the standard fields of an issue. You can synchronize it as any other issue fields.

This article provides description and scripts that help you to synchronize issue watchers and create new users if they do not exist on the destination side.

The Exalate has a helper method to create users with no permissions. Use this method to sync the watchers list.

More details about watchers issue field.

Source side

Add the code below to the Outgoing sync (data filter) to send watchers 

 replica.watchers = issue.watchers

Destination side  

Add the code below to the Incoming sync(create/change processor) to create users if they don't exist on the receiving side.

issue.watchers = replica.watchers.collect { w ->
    def _localW = nodeHelper.getUserByEmail(w.email)
    if (_localW == null) {
        // We create a user with no permissions, and with password equal to the username
        return nodeHelper.createUser(w.username, w.username, w.email, w.displayName)

Updating watchers on Jira Cloud does not trigger Exalate synchronization event. It means that watchers will sync with the next issue update like new comment, description or other fields update.