Arse. While traveling recently I used the stock Google Camera app on my phone to take photos. I presumed they'd all be geotagged, and I could load em up in JOSM and add to #OpenStreetMap the things I took photos of.
But it only geotagged a few of them. 😢
Another option is taking a picture with #OsmAnd.
Finally, Tibor Kaputa's camera application is lighting fast, but I haven't checked if it geotags.
Camera (A camera with flash, zoom and no ads.) - https://f-droid.org/app/com.simplemobiletools.camera
Then on the map you click either on the point you want to associate a picture with our on your own position marker, go Actions → Take photo.
Downloading seems to be possible only by selecting the pics one by one (tap on) and using the share action.
I have on occasion used it in conjunction with OSM notes.