According to http://appamatix.com/delete-snapchat-messages-conversations
If you're interested in deleting the Snapchat messages you either sent or received while using the app all you need to do is a few steps and you'll be saved from a potentially embarrassing moment that no one wants to be stuck in.
Follow these steps to delete the messages permanently on your phone. Keep in mind that this won't delete the messages on someone else's phone it will only be on yours. This is because Snapchat downloads all messages on people's phone and not on the Snapchat server. This is still a lifesaver, though for when you are suddenly aware of people looking over your shoulder (yes, I do frequently take public transit, why do you ask?) or else you are nervous about somebody getting their physical hands onto your physical phone.
- Open the Snapchat app and you'll see a little Snapchat ghost at the top center.
- Tap on the ghost and you'll see your Snapchat icon with your name and score. Tap on the gear icon at the top left to get into the settings.
- From there, scroll down to Account Actions and tap on "Clear Conversations" in that menu.
- You will see a list of all of your messages with an x right next to it. You will get a message that says "Are you sure you want to clear this conversation?" and you can get the option to clear and clear and don't ask again.
- You could alternatively choose to clear all of your messages at once if you don't want to have anyone see your messages at all.
NOTE: When you clear the conversation on your feed, it won't delete any saved or sent messages in your conversation. You can delete your saved messages as well, however you can't delete messages that have been sent to someone else.