Here's a first responder with some input.
My crews recently discovered my feed (it wasn't put up for their benefit, it's something I'm doing as a hobby) and they love it. They use it for listening to on-scene comms when they are out of range of our radio system. The only thing they dislike is the fact that it has a delay (inherent in the design of the feed and the Internet). In other words, they don't think there's a need for a delay, nor do they feel that there is any safety concern with having our dispatch-related audio and wide-area comms audio on the Internet. (Our simplex fireground channels are not streamed, more because it is impractical to do so due to the low power nature of them and the impossibility of getting a consistently strong signal for them due to our huge area.)
The other thing that talks not just to the discussion of the feed, but about the sensitivity about emergency comms (or lack thereof), is that my department is rather open about our responses. We have a Twitter feed which I manually update with calls we are responding on as it happens. Naturally we don't say "We're going to 5432 1st Street apartment 203 to deal with 87 year old Mr. McGillicutty who fell off the toilet", we say "We are responding to the downtown area for a medical assist". Media outlets follow this Twitter account and pay close attention to what is going on. With this, we have a great relationship with a lot of agencies and are getting decent publicity out of it. There's no reason to hide yourself away from the public. Things which are private can be discussed in private - either face-to-face or off the monitorable airwaves. (Anything we wouldn't say shouted from the top of our rescue truck through a megaphone is never broadcast over a radio - instead is communicated over a less-easily monitorable network such as cellular telephone.)
There's nothing inherently unsafe about having your comms broadcast on the Internet (and via smartphone scanner apps, and etc). If you are communicating things on those frequencies that have the potential to jeopardize the safety of your responders, you need to re-evaluate (a) whether those things need to be said in that "forum" and (2) whether or not you should have a more secure "forum" for saying those things. Safety is not compromised by the world knowing that your crews are responding to the intersection of Walk and Don't Walk for a car crash. Neither is privacy of the patients - they're in a public place. People could just as easily tuck in behind the engine as it goes by and follow them out of curiosity, rather than hear it on the scanner, and they'd still get to the scene.
If you (as a radio system operator) have police operations that specifically involve communications to circumvent the movements and/or operations of the bad guys, i.e. gang unit or drug team surveillance, or even tactical/SWAT team positioning on a barricaded subject, it is your responsibility to ensure that these communications are protected from "prying ears". This however does not include general patrol. There is nothing inherently sensitive about routine police work i.e. that you are sending a car to 5678 Xyz St. for a domestic dispute or to the Downtown 7-11 to investigate a shoplifting (or even an armed robbery). It should not be the responsibility of an audio streaming service to obfuscate any of these comms - through delays or other measures - to prevent "the bad guys" from using it to their advantage. A police or public safety department finding itself having such issues needs to evaluate, as I said earlier, what it says on the radio, and whether or not that part of its radio service needs to be secured (if that stuff really has to be broadcast anyway).