Here are two descriptions of how the initial charge difference occurs:
"Although water is full of charged particles - ions from dissolved salts and from the breakdown of water itself, the water that initially falls through the holes in the bucket will be uncharged on the average. However, the randomness of the universe will soon cause there to be a slight charge on one the can/wire systems. Perhaps a drop will fall that is slightly charged, or maybe a cosmic ray will hit the wire and cause a small charge to form on one of the little cans.
Let's say that the little can on the left (and hence the large can on the right) is slightly positive. Then, this positive charge will have three effects on the falling streams.
1) Negative ions will be slightly attracted towards the left most hole, so that the water falling through it will, on average, be slightly negatively charged. Thus, the left most large can (and the right side small can) will become negatively charged.
2) Conversely, positive ions will move to the vicinity of the right hand side whole, making the right hand side stream positively charged.
3) The positive charge on the can will repel any slightly positively charged drops on the left hand side - this will tend to increase the total negative charge falling into the left hand large can."
http://outreach.phys.uh.edu/electexperiments.htm
"SELF-STARTING
But where does the first charge come from? In fact, if you build such a device, it will usually create voltage all by itself, spontaneously, without being pre-charged. During dry conditions everything near the generator ends up with a tiny electric charge just from being handled. If one of the upper cans is slightly negative, it will cause the water to have imbalanced positive, which will start up the other side of the generator, which will make the charge on the negative side become larger, etc., over and over.
It's like balancing a penny on edge: it's hard to start out with a perfect balance, and usually it falls one way or the other. Same with this generator. If there's a tiny electrical imbalance at the start, the generator will amplify it over and over, and the voltage will "fall over" to either one polarity or the other. A high voltage will magically appear from nowhere. (But nobody knows which side will start out positive and which will be negative.)"
http://amasci.com/emotor/kelvin.html