OpenStreetMap did not get its initial data from MapQuest or any other map providers. There have been some imports from free sources like the TIGER data in USA and AND data in the netherlands. Netherlands. Common errors can be from such an import.
Other maps can freely import data from osm as long as they give attribution and license the result under CC BY-SA. This far there are [MapQuest Open][1] and [Bing Mapnik layer][2] that both use only osm data. We hope that the comming coming license change would make the map data more attractive to map consumers. You can also find osm data used in wikipedia, forsquere, geohashing, apples Wikipedia, Forsquare, geocaching.com, Apple's iPhoto and many other places.
We are still waiting for Google to come with a osm layer.
[1]: http://open.mapquest.co.uk/
[2]: http://www.bing.com/maps/explore/#5003/s=w/5872/style=Mapnik&pid=50735