Nearly all municipalities have laws about keeping vegetation from growing out into paths. Furthermore, there are often stricter laws about anything (whether plants or not) that poses a health risk.
I always think it is best to try approaching the property owner first. If I were you, I'd knock on the person's door or leave a note, explaining the situation. If the person doesn't want to remove them, or perhaps doesn't have the resources to do so (such as an elderly person who doesn't have much income nor great physical health) you could offer to remove them yourself.
If the person refuses, I would then contact the municipal government (depending on where you live, this could be a city/town, a township, a county, or some other unit of government.) They may threaten the person with a steep fine, and hopefully that would get the problem addressed. However, some governments are more proactive about this stuff and some people may continue to refuse to comply.
In this case if it dragged out this far, I would just go and remove the stuff yourself. Cover up with glove and long-sleeves, and you then have two options: either just clip the nettles back to the edge of the path, or for a more aggressive control solution you could try uprooting them. But I think, since it's not your land, I would recommend just clipping them back.
I would not be secretive about doing this. If you are confronted by the landowner, or by police, you could explain to them that you are protecting the health and safety of your children who are using the public right-of-way. You are not going to get arrested for doing something like this, especially if you're staying on the public right-of-way and being restrained about how far you go back. Anyone seeing what is going on is going to understand that you are the reasonable party here.
Good luck! Hopefully just talking to the person works!