If I an doing analogRead, aren't I reading how much current goes into the pin?
No you are not. You are measuring the voltage on that pin. The current through the resistor to ground produces a voltage across the resistor, that is what you measure. The higher are the resistances the more voltage is produced for a given current.
f 1 Mohm is fine for 1 piezo, why is it too much for multiple piezos
You have been told this already. It is because you have to charge the input capacitor on the sample an hold circuit from this voltage.
There is only one A/D converter and there is a sample and hold circuit on the front end. In front of that is a switch which switches through the voltage from one of the six analogue inputs.
With only one sensor the voltage never gets switched and so the capacitor voltage always tracks what is on the input. Once you put more than one sensor on and switch between the two you need a lower impedance to charge the capacitor quicker, or you need longer for that capacitor to reach the input voltage, hence a double read perhaps even with a delay in between.