As per my knowledge, messages cannot be saved in snapchat. If you see any messages there, they have been saved (they will be in bold). If they don't un-bold when you tap on them, the other person has saved the message, and you are not able to delete it (you'll have to ask them to un-save it).
Hereof, how do you know if someone saves your picture on Snapchat?
Look for messages with grey backgrounds. If you see a message with a grey background, it has been saved by either you or your contact. Messages you save will have a vertical red bar to the left of them, while messages saved by friends have a blue bar next to them. You can save a chat message by tapping and holding it.
How do you delete conversations on Snapchat that the other person saved?
Tap the message to unsave it on your side. If you also use the Snapchat account on other devices, make sure that you unsave the message on the device. Ask the other person to unsave the message. If they refuse, there are Snapchat history erasers that claim to be able to delete messages that someone else saved.
Can you tell if someone saves your picture on Instagram?
The selection of saved Instagram photos can be accessed from the user's main profile page, and when enough are collected, the saved feed looks just like a user's personal feed of photos. (However, photos that users save will be private.) Users don't receive a notification if their photo has been saved by another user.