I don't want to use WMS layers, but to actually store the data in my local server, where can I find street data from the world?
OpenStreetMap is probably your best option. Cloudmade, a company that makes tools and APIs for OSM, provides regular data extracts for free on their website in many different formats. The only downside is that the extract geographic area can be quite small (the size of states in the US) so that may not work if you need world data out of the box.
If you truly need the world, you'll probably want to download the OSM planet file. This contains all of the OSM data for the planet and there are a number of tools for importing it. I often use osm2pgsql.
http://www.openstreetmap.org/ Check it out.
For the United States, there is always Tiger data from the Census.