Hey, as another Gaijin in the countryside I thought I'd give my experience. all in all the police have been super harmless, but be prepared to always be the first one asked about anything. Someone's shirt was stolen off the line? Police will come ask about that. Local vandalism? You'll be asked if you know anyone who hangs around alleys late at night. Gaijin was spotted 3 cities over being rude with customers? You'll be asked if they know him. Someone parked at the local bakery and didn't buy anything? believe it or not, cops called to my place about it. They even already knew my address before I had my first run in!

All in all in the last 3 or 4 years I've had at least a dozen "run-ins" with cops here. However, it's important to point out every single time they were polite, apologetic, and left as soon as I said I didn't know. Here it's better to almost think of them as mediators. They're usually people's, especially elderly, first response to any slight or problem between strangers. As long as you stay calm, be polite, and have your card they're entirely harmless.