The EGNOS and WAAS ground stations provide correction information to the transponders on the geostationary satellites. You don't receive the ground station information directly.
Not according to
About EGNOS | EGNOS Portal . There's a useful videoclip on that site which explains how EGNOS works. Ground infrastructure receives normal GPS signals which are then checked and any error corrections are sent to the geostationary satellites (Inmarsat and Artemis, soon to be replaced by new Astra satellites). Those satellites then send corrective messages to EGNOS-enabled receivers.
My (wonderful) Garmin had EGNOS reception selected ON; today I'll try ith with EGNOS reception selected OFF and see whether I get similar accuracy.
GPS satellites are in lower orbit (20000 km) than geostationary (36000 km). Given that many vehicles in the GPS constellation were in orbit a long time before EGNOS, I can't really see that a direct satellite-to-satellite link would be feasible - or that the US would welcome other nations' systems doing so.
Anyway, 'tis all magic and beats the pants off ADF!