If you want to delete text messages on your iPhone for both sides, there are a few different ways to go about it. The first way is to go to your Messages app and select the conversation you want to delete. Then, tap on the three lines in the top left corner of the screen and select Delete Message. If you want to delete a specific message from a conversation, you can do that by selecting the message in question and tapping on the three lines in the top left corner of the screen. Then, tap on Delete Message. The last way to delete text messages on your iPhone is to go to Settings and then Messages. Under Messages, tap on each individual conversation and then tap on Delete Message. ..