Seems not possible with the JIRA clone feature due to https://jira.atlassian.com/browse/JRA-28038
However you might be able to achieve that using an add-on like this one https://marketplace.atlassian.com/plugins/org.swift.jira.clone-plusand the only downside is that it is paid :)
Also you can write a listener with script runner that will fire on createIssue event and check if the issue is cloned and copy all the watchers from the original issue.
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