When screenshot notifications happen
Snapchat can notify users about screenshots taken inside signed-in app experiences, depending on the content type and current app behavior.
Why public browser downloads are different
A browser download of public media does not use your Snapchat account, so the app does not have a signed-in account identity to attach to a screenshot or download action.
Private stories and chats
Private stories, chats and direct snaps still happen inside Snapchat's logged-in environment. Screenshots there can be visible to the other person.
For public media, use the Snapchat story downloader. For notification basics, read does Snapchat notify story views?