I’m trying to keep an eye on my partner’s Snapchat activity, but I don’t want them to know I’m screenshotting their stories or chats. Does the app actually send a notification for either of those?
Yeah, Snapchat does notify for screenshots in chats and snaps - you’ll see a little screenshot icon show up in the conversation. Stories are a bit more forgiving though, no notification there.
But honestly, if you’re trying to secretly screenshot your partner’s phone, that means you either have their phone unlocked in your hand already, or you’re logged into their account - both of which are a bit of a different conversation than the app itself. 
Might be worth just having a direct chat with them - way less stressful than playing screenshot ninja.