Your diagram uses two discreet 12 volt transformers with two discreet bridge rectifiers and two discreet filter capacitors. That is not what you will get with the transformer you purchased. That would be similar to the discreet circuit I posted after the split +/- supply version. I don't know why it was done using two transformers? Where is the rest of the link for that circuit you posted?
https://www.royalrife.com/em8v.html These are the construction notes. I am using a 10,000uf 50v filter cap and the resistor is 68 ohm instead of 680. The designer told me I could use these 24v 10A transformers and that is also what the construction notes say. Maybe he wants me to just use the 12vdc side?
From the picture of the label i posted Is this correct? One channel black gets 120 and red gets neutral, the other channel the red gets 120 and the black gets neutral. Maybe I just use the +12vdc side of the output split between the 2 circuits. It is 10A and that is more than enough to run both circuits.
As to the transformer you connect the two red primary wires together, you connect the two black primary wires together. Red to red and black to black, got it? Now it matters not how you connect the 120 mains. You can connect mains hot to either red or black and mains neutral to the remaining pair.
Now from reading this:
(B) 120 V to 12 V Transformers -- Marlin P. Jones (800-652-6733) has the 4 amp 12 volt (6-0-6) stock number 7840-TR. They also stock the 10 amp
12-0-12 stock number 7846-TR. The 10 amp model is much larger and heavier and requires a larger box. The 5 amp TX-125 from All Electronics works well
My guess is you only need a single 12 volt source. Using 1/2 of your transformer will give you 12 volts @ 5 amps which from what I read above should be adequate and work. Therefore on your transformer secondary connect one blue line to the AC in on your bridge, matters not which blue you use. Connect the yellow C/T line to the other AC in on your bridge. Place your filter cap across the + and - DC out of the bridge. That is your 12 volt supply. The remaining secondary blue line is not used.
I suppose it depends on what's on the other end of those outputs, right? If the two circuits were completely independent, then I'm sure this would work. Not sure what would happen if you tied the negative outputs together into a common ground, though.