So like the people already said.
When you rectify the AC voltage to DC you multiply by sqrt(2) ... so if you have a 230v to 12 VAC transformer rated at 12 W , it means the transformer can do 12 VAC at 1A. But the DC voltage after rectification is ideally 12v x 1.41 = 16.92v . In reality, as the current goes through 2 diodes at any moment, you have another drop of about 0.5-0.8v per diode.
Therefore, you have now about 16v output when you have no load on the transformer. When you have some load, you will have about 15.5v output.
But the transformer is still rated for 12 W, so now you have 15.5v at some amps = 12W ... the current is 12 w / 15.5 = 0.774 A ... so now you only have about 700 mA at 15.5v
The linear regulators have also a certain efficiency, going to 9v from 15.5 would be done at around 85% efficiency if you use a plain lm317 or something similar to the classic 7805, so you're actually looking at generating about 9v @ 600 mA from a 12 VAC 12 W transformer.
You should pick a 8-10 VAC 1.5-2A ( so about 15-20 VA minimum) transformer, for example something i found very fast :
http://www.digikey.com/product-detail/en/VPS16-1600/237-1260-ND/666146It's 16 VAC 1.6A or 8 VAC 3.2 A
8 VAC @ 3.2 A -> rectifies to 11,3v -> drop ~ 1-1.5v to the couple of diodes -> you're left with about 9.5-10v at about 2-2.5A with a decent smoothing capacitor (I'm thinking 1000-3300uF)...
Now you can use one or a couple of low drop linear regulator like MIC2941 (
http://www.digikey.com/product-detail/en/MIC2941AWT/576-1138-ND/771607) which needs at most 0.35v drop to generate 9v @ 1A so as long as the input voltage won't drop below about 9.4v you will be good.
Since you probably won't max out both outputs at 1A, you could use such transformer that can give you max 9.510V DC @2-2.5 A and two linear regulators to get 2 x 9V DC @ 1-1.25A each
ps. You don't need to pick one that has such low margin for the regulators, you can for example pick one with 10vac 2.5a-3a for about the same price:
http://www.digikey.com/product-detail/en/VPS20-1250/237-1264-ND/666150http://www.digikey.com/product-detail/en/VPP20-1500/237-1077-ND/242521This will give you a bit more leeway for the voltage drop.