When you create your display name it is stored on the Golden Frog servers. When you have a conversation with any friend for the first time, Cyphr shows your display name to your friend and stores your display name locally on your friend's device. If you change your display name after messaging with a friend, your friend will not see your new display name. Additionally, any changes that your friend makes to your display name will not be visible to you.
If you need any further assistance, please contact support and we'll be happy to help!