Actually, with the LocalGsmNlpBackend you can download and save data from MLS (and OpenCellID) on your phone.
You’ll need to install µg UnifiedNlp. Here’s a guide how to.
Edit: I also use AIMSICD (OpenCellID, no ranks yet), OpenWLANMap (rank 224), Radiobeacon (openbmap, rank 22), Sensorium (Open3GMap) and Wigle (found out it’s data is not free at all) to contribute data to similiar services, but I like Mozilla Stumbler (rank 691) the most because the app is beautifully designed and easy to use. What would be cool though is to have an app like Mozilla Stumbler that contributes to all those services at once. (or the ones you choose)