For Android users, Video Downloader for Social Media - No Watermark by Avalon Inc is available to download through the Google Play Store, while iPhone users might want to consider something like the Shortcuts app, available through the App Store. If the creator has not enabled video-saving, a third-party app can be used to save the video. Alternatively, in the share menu where the URL can be copied, on the bottom row, there is also an option to save the video. One of these options will be save video. Tap that option and the video will download to your device's storage.
To save a video, simply tap and hold the screen and a pop-up with options will appear. For those who want an actual copy of the video to send, share, or keep for themselves, the process is also fairly simple, assuming the video's creator has enabled saving in the first place.