Once messages are set to Active and have begun to send, you should be able to verify that they are being sent from your Sent Messages tab. You can choose to view everything being sent (the default view), everything being sent sorted by template name (you can make your selection from the drop down menu), everything sent by marketplace, or by what each buyer received by searching by the Order-ID or buyer email address.
You should then also be able to verify that the messages are being sent to your customers by going in to Seller Central's Buyer/Seller Messenger and checking the "Sent" tab. That will include custom messages sent by the Communications Center through Amazon to your customers. You will not see Amazon's Request Review messages here.
Not seeing your sent messages in Seller Central? Make sure your "Sending Email" email address is set as an Approved Sender. Learn more about the "Sending Email" email address here.