They could easily argue that they are no more intrusive, annoying and harassing than a Mister Softee guy who follows a group of kids.
Re: The NYPD. I don't know if you have ever witnessed an incident wherein Shomrim are present before the police; I have seen the arriving NYPD officers greet the Shomrim volunteers at the scene by name.
If you want to test their response time, you could probably just walk around some of the residential streets singing at the top of your lungs, dressed in your present attire. I suspect that alone would get their attention, and cause them to follow you.
Pro tip: Don't act so oddly you meet the definition of an Emotionally Disturbed Person.
Let me know how it goes. Like I said above, why use volunteers to follow people when you can use cameras?
Cameras are so much better..... they can provide the same (or better) level of surveillance, without the -um- "political" risks.
If you play your cards right, you might even get government to pay for 'em... Of course, you have something to exchange.