• All data is uploaded anonymously, so I can't download or save the raw data I've collected. This means I can't look at it except aggregated together with other users' data and fuzzed on a map Mozilla provides, I can't play with it, I can't upload it to other similar projects, etc.
• In fact, nobody can download a dump of the data because of privacy concerns for data contributors and access point operators. At best, you can consume a derivative of the data by using the location API.
• In an attempt to protect the privacy of data contributors, Mozilla "fuzzes" acquired data before displaying it on their map. Unfortunately, they've fuzzed the data to the point that I can't tell which streets I've already been on and which I haven't. (Half of the fun of this was seeing things on streets I'd never been on before.)
It's mildly tempting to fork MozStumbler and have it also upload the data I collect to a server I control, just so I have a copy of my data. (In particular, it might be fun to throw together some code for algorithmically locating cell sites and access points, just to see how well I can make it work.)
You could use WiGLE's client if you want that; the client logs to a local file in addition to (optionally) uploading to the main database: https://wigle.net/
OpenStreetMap has lots of opportunities for this kind of thing, it's one of the reasons I contribute a little.