Existing verizon.net email users have two options:
- Continue to use their verizon.net email address, but it will be hosted by AOL Mail.
- Give up their verizon.net email address and find a new email/webmail provider.
From a deliverability perspective, I think it is safe to assume that most of the Verizon email infrastructure is being retired, and that the verizon.net email address domain lives on a just another of the AOL email domains (like cs.com, wmconnect.com, aim.com, wow.com, etc.) There likely won't be any difference in reputation or filtering systems between verizon.net and aol.com mailboxes, when the transition is fully complete. This could be subject to change, so stay tuned.
Update: Laura Atkins of Word to the Wise rightly points out that there's no timeline or deadline published anywhere in this Verizon notice. When exactly will Verizon shut it all down? We shall see.
June 26, 2017 Update: As related to me and others by AOL, the Verizon mailboxes that remain have now been transitioned to AOL's mail servers.