For the requests I have from our management team we will be creating a few custom timesheets, being able to share them will be a huge benefit so that each user does not have to re-create them or update them if needed. Voted for this improvement.
As a workaround for until this issue is fixed you can share timesheet URLs. You can copy the url and send it to another person. If he is already logged in to JIRA, the custom timesheet will be automatically loaded with the same settings.