And another way of looking at the river problem, what if the farmer used nitrogen fertilizer first, can someone new moving next to the downstream river then have a right to force them to change fertilizers? And the answer is no, but with property rights things can be changed to get a more efficient outcome, for example someone doesn't want dead fish passing through their river and the farmer doesn't want to lose profits changing fertilizer and being less competitive, the neighbor can pay the farmer to stop doing things that kill the fish. Or buy the land from the farmer, or pay the farmer to move. Or buy technology to clean the river.
Someone might wonder why those who want a clean river must pay, but put the other way around, why do people who buy goods from the farmer must pay for a clean river? There needs to be some balance between these two groups somewhere. It could be the farmer is willing to pay those downstream to be able to pollute their rivers.