Your radiator has an internal cooler, that is made to withstand the pressure of the transmission lines. It's not that uncommon for the trans cooler to have a pinhole leak in it.
If it is your trans cooler leaking you will very rarely get coolant in your trans fluid. The only way to fix an internal cooler is to replace the rad (on copper/brass setups you can remove the tank & replace the cooler) or just run with an external cooler only. If you want to test the cooler (the back yard mechanic way), pull the trans lines off, blow the fluid out of the cooler using an air compressor, plug one of the lines on the rad, put an air line on the other trans fitting (on the rad), provide about 40lbs constant air pressure, and look into the filler neck for bubbles. Normally it is just a "fizzer" caused by a pinhole leak. Do this with the vehicle NOT running. Sometimes it takes a couple of minutes to see the leak.
P.S.
Once you have fixed the trans fluid problem, have your cooling system flushed with chemical and I don't recommend just using a garden hose attachment. Go to a radiator shop and have them flush it with pressure. I used to work in a rad shop (for about 5 years), and have dealt with stuff like this lots. The garden hose thing is just good for a rinse.