Ok, so I guess this is where using the water analogy breaks down, right? With the water analog it's easy to have one water reservoir be filled while another water reservoir is empty. This obvious and sometimes massive imbalance of water (say, water flowing from a lake into an empty hole dug into the ground) is common to have.
But electrons don't behave that way, or rather, it's very very hard to get them to behave this way because electron imbalances are very difficult to maintain. You can have a tub filled with electrons and another tub empty of electrons and when you link the two you will have current that's non-circuitous, but it's very difficult to create this.
So speaking very technically, a complete circuit is not strictly necessary for electron flow, but realistically it is necessary due to the fact that electron imbalances are extremely difficult to maintain?