This isn't possible without third party data. Things like the refractive index of the atmosphere (which depends on weather patterns), and the fact orbits are slightly warped due to varying gravitational fields and radiation pressure from the sun, all add up to location innacuracies unless you can correct for these effects.

That correction requires data, which needs to come from a third party because the satellites don't transmit it.

Correct, if only L1 is used. However as GPS exists today receivers can track phase variations between L1 and L2 without the codes necessary to decode the signal on L2.

Actually its a bit more sophisticated. Galileo ground stations measure their reported position against the actual, and then transmit correction data to the satellites which then retransmit it with the rest of the position data.

Groundbased gps transmitters might help to increase accuracy.