I'll throw my $.02 in. I don't see a problem with public-service departments going out and running brief errands in their response area-- because they are also familiarizing themselves with community buildings, citizens, etc, and unlike police we aren't expected to be proactive in seeking calls. As long as the above doesn't put gross wear and tear on the vehicles and you remain ready to take calls.
I work for a private service-- so what I do with my time and vehicle and between me and my supervisors/company owner, generally not the public unless we're disrupting them. I'll admit it, I went to Target yesterday for 10 minutes on a 12/24 hour shift, then got lunch at Panera. I parked the truck in the parking lot about 15 spaces from the store, and left my partner in the truck (with it on, because it was effing cold and company policy requires it to be on for Rx temps). It's simply not practical for either me or my partner to drag the drug box through Target with us so the truck could be off.
In this case, we weren't working a 911 contract, but an ALS transfer/SCT truck. We do some STAT transfers, but were less than a mile from our contract hospital and could get there quickly if we needed.
South Bay Center...good times.
At my hometown volunteer service, there are few rules (don't even get me started), and we are allowed to be anywhere in our area and can generally be on the other side of our borders provided we stay close. We can go further with officer approval, but otherwise don't have to notify our four dispatch agencies, as they have no expectation of knowing where we are, nor do they care (again, don't me started).
At my campus service, we don't post in quarters, and can be anywhere in the response area. It's best to stay close to the center of campus, but otherwise we're free to do as we please.
At my commercial IFT job, we get assigned posts in general geographic areas, and can be anywhere tangentially close. If we need to step out, we can do so en route to post or once we get there. We can also make diversions if there is somewhere we need to go by calling dispatch and asking (they almost always say yes). We just call over the air to say we'll be out of the truck (we don't carry portables), and call back in when we return. Dispatch has our phone number and can reach us if they need us.