If you have efiled both federal and state returns, then there is no need to mail in copies, however it is always a good idea to print copies for your records.
If you have "heard" different, please know you should contact the State office directly and ask clarifying questions. Here is a link that can assist you with that, https://ttlc.intuit.com/community/refund-status/help/how-do-i-track-my-illinois-tax-refund/00/26562
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