If you keep an eye out on nasioc you can often find sets for <$600 with wheels and slightly used winter tires already mounted. You can't be that picky about how they look if you're trying to save that much money but they are often totally functional solutions.
Remember, AWD does nothing when your foot isn't on the gas pedal. In other words, it doesn't help you stop. Tires help EVERYTHING about your car. You can't run summer tires in the winter no matter how good the traction of the car is, the tire compound is all wrong. And worse, if you run into snow, you're in for real trouble. They'll quickly suck up some of the moisture and turn into hockey pucks rather than tires.
And yes, it's kind of odd to have an AWD, all season friendly vehicle sold with just summer tires. 09 was the first year they did it. They were getting pissed off at car magazines (C&D, etc) for comparing their WRX to the mazdaspeed3 and similar while leaving the all season tires on the WRX. It's "competition" in these types of articles (usually FWD monsters with no grip) always had summer rubber so they just caved and put summer tires on the WRX so it could compete properly.
2015 STi: Still Stock
04 STi, 04 WRX: SOLD