It seems to me that the current of the charging circuit would gradually decrease as the battery reaches full charge. My initial observations done by plugging an ammeter into a charging circuit seem to support this. Is this generally correct? Is this how an automatic trickle charger "knows" to stop charging a wet cell battery? Or does it instead go by the voltage of the charging circuit?
Well, basically at any given moment, the battery acts like it has a secret, internal open circuit voltage (Voc) and a series resistance (Rs). The charger outputs Vin. Then the charge current, Ic, is: Ic = (Vin-Voc)/Rs. As the battery gets charged, Voc goes up, so Ic goes down. It may well be that Voc is effected by other things besides state of charge. But this is an OK first approximation to help you understand how the current is set.
Your observations are correct.
It does not 'go by the voltage of the charging circuit'. Instead, the amount of current transferred - and the direction the current flows - is largely determined by the voltage difference between the two sources, and the resistance between them.
In general, your trickle charger does not 'know' to stop charging the battery at any given point in time.
Think of it more like this: The trickle charger has a constant output at a given voltage (Think of this as Vin in mkeith's answer's example). This will always be true.
Your battery, at the time you begin charging it, has a specific voltage - in @mkeith's example, Voc.
As this battery is plugged into the trickle charger, the difference in voltage, along with the resistance of the connection will determine the amount of current that's transferred to the battery.
This resistance will likely stay the same as the battery is charged; however, as the battery gains charge, the difference between the battery and the trickle charger's independent voltages will decrease.
If you're familiar with Ohm's Law, you'll then understand that as a direct result of this, the amount of current that transfers will drop.