No, Snapchat and all other social networks are inherently not private. And the fact that Snapchat promises to erase your photos lulls many users (especially teens) into a false sense of privacy.
In addition to the ability for the recipient to take a screenshot of a Snap, there are also third party apps that can capture Snaps. This week saw one of those third-party sites hacked, and 100,000s of photos were stolen. It is likely that this collection will be published soon.
Although it was not Snapchat’s servers themselves that were hacked, Snapchat should take some responsibility for the ease with which these third-party apps can access Snaps from one’s phone.
For details on this recent hack, read this CNN article: Snapchat isn’t Private..