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So after some research I decided to stick with the OSM metadata. I found the Overpass API, which is basically a query language, which I managed to understand reading this: http://wiki.openstreetmap.org/wiki/Overpass_API/Overpass_QL and testing it here http://overpass-api.de/query_form.html I ended up with this query: way ...
Google maps roads API can do that, if a non-OSM solution is ok with you. See snapToRoad method which returns the GPS points snapped to the nearest Road. You can take the place id from the response and use Google Maps Places API to get corresponding Road name. https://developers.google.com/maps/documentation/roads/snap
I think JasonT is on the right track (excuse the pun). Note, however, it is the concept of a buffer that you want, not to actually create a buffer. If you were using PostGIS (or similar software/libraries), you'd use ST_DWithin() to reject public road points that are too close to your GPS points.
You could try the plugin Qsphere... its version 2.6.5 is very stable and works fine.
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