The whole purpose of Snapchat is to be able to send a photo that disappears after a given amount of time. Totally like Mission Impossible, only you don't get that cool voice saying "this message will self destruct in 10 seconds." Users love it mostly because they want to send 1000 photos a day of goofy faces or their cat, and don't actually want to keep them all. That said, disappearing messages are still used for pictures or video you really do want seen and thrown away.
To prevent recipients from screen capturing these (supposedly) ephemeral images, Snapchat introduced a notification to show when another user has captured your image. Recently, Instagram introduced the same on their disappearing messages. Here’s the thing, like all tech, there is a way around it.
Open your Instagram or Snapchat app and find the photo you want so it loads to your device. Do not open it yet, this is what fires the notification process.
Turn off your WiFi and cellular transmission. Easiest way to do this is via Airplane Mode.
Now you can open the photo and take your screenshot.
This is key. You cannot just close it. You need to actually quit the app so that the app kills off all processes it has started with your device. Including that notification.
Turn your connections back on and then you can reopen your app.
If any of this is too complicated you can always use a second device to take a picture or video. Or, if you have a connection to a second screen like a laptop, you can screen capture from there.
PS All things online should be treated as Public and Permanent. No matter how private they start out, digital media has a way of finding its way in to the wrong hands by intention or by accident.