Whether your iPhone permanently stores the messages you read on it depends on which app you use to access email. IOS 7 brought a change to the way emails are downloaded and kept on your iPhone.
With earlier editions of iOS 7, the Mail app that ships with the phone kept only a user-defined number of messages on the phone at any one time. Beginning with iOS 7, the iPhone keeps all emails from your mail server in the Mail app, which can result in an enormous number of emails being downloaded and stored on the device. These can include emails you have previously accessed using another device. Because there is no easy way to delete a large number of emails, some users are unhappy with the change in Mail settings.
Mail App Workaround
Depending on your email provider, you may be able to stop the Mail app from storing all messages if you delete and re-add your email account. Under iOS 7, the set-up process usually lets you decide how many days to sync your account. This means older emails are removed from your iPhone automatically but remain accessible on your account so you can access them on other devices or redownload them to your iPhone later. If you don't see this option during the set-up process, this workaround isn't available with your email provider.
Other Mail Apps
Whether other mail apps for the iPhone store messages on the phone itself depends on the service provider. To check an app, access and read several new messages. Close the app and put the iPhone into Airplane mode by swiping up from the bottom of the screen and then tapping the airplane icon. While in airplane mode, check whether you can read the messages in the app. If so, they are stored on the iPhone.