As stated 18 days ago, I took a break from trying to make an Arduino board talk to an ESP8266 WiFi module. In the meantime I have both updated the 8266 firmware and received level converters.
Nothing changed: still no data flow from Arduino to ESP8266 over the hardware UART.
After three nights spent debugging software and hardware, trying to understand why both of them could happily speak to a computer, one at a time, but not to each other, I took the last chance: I fed directly the Arduino output at 5V/TTL. Make it or (literally) break it.
Well, it simply did it.
My ESP8266 specimens seem to be a "deaf" at 3.3 V, but they happily accept data at 5 V (datasheet says inputs are 5 V tolerant). ESP8266 power supply remains at 3.3 Volts.
More than ever, YMMV (Your Modules May Vary). I am not suggesting this procedure as standard practice. I report it worked for my specimens.