Android
Files are stored in the MapsWithMe folder (in the root of the internal storage of your device) or on SD card in Android/data/com.mapswithme.maps.pro/files/MapsWithMe folder, if you store downloaded maps on SD.
Maps are stored in a subfolder with map data version in its name, for example, 200920. Bookmarks are stored in a bookmarks subfolder and have .kmb extension*, for example, My places.kmb. You can access the folders with the help of almost any file manager like Samsung file manager or Total Commander.
*Note: if you copy bookmarks directly from the app folder they may not work on other devices, instead you should export bookmarks in KMZ (KMZ is a zipped KML) format directly via the app.
iOS
You can access the app files with iTunes: connect your device to a computer → open iTunes → select your device → File Sharing → select maps.me.
On MacOS Catalina and newer connect your device to a computer → open Finder → choose connected device in the sidebar of the Finder window → open Files tab → select maps.me.
