Running water at best would encourage settlement not compaction. Worst case would be erosion. To use water you would have to put it in a flexible container and wait for months. The same applies to any weight. You have to wait for the soil particles to shift with temperature and moisture changes to lead to compaction. This method is used in construction on thick layers of poor compaction over large areas.

One alternative is hiring a paving company to come in with a small steel roller.
Another for an already paved area where you don't care about the bricks is to drive a car back and forth. Be sure to cover the whole area with the tires. Drive slow and make gentle curves.

Have you checked the actual cost of renting a machine? It might be worth renting one for the peace of mind when you know a good job has been done.

Get an 8' 4x4 and by lifting and dropping it onto the area to be compacted you'll achieve good compaction. However, after 2-4 hours of this, plus possible pain and misery, plus visits to a chiropractor you may weigh the costs of doing the job this way with renting a machine.

--Bill