Well, with any diverse group of employee types (Type A, laid back and so on), how the police feel about the opinions and feelings toward police is a varied answer.

Some care and some do not (and those that don't care also have varying behavior from just a tepid response to those that have a "FUCK them" perspective.

In my experience what I felt was the most common response was that of a feeling of an "us versus them" belief.  And what this is simply based on is the type of encounters the police have with citizens.  By this I mean, one of the largest groups of policing calls are of negative encounters - family disputes, fight calls, bad things that have happened calls and so on, so it is easy for police officers to believe that the overwhelming population is negative toward the police, when that is really not true.  Many of the police encounters are negative with a small, often repeated, "customer base." 

I used to tell those I worked with that the overwhelming populace is satisfied with the police but we don't encounter those people the most often - we most often encounter people in a crisis and sadly, those calls seem to take precedence in the thoughts of police.

But many of us would just see things as coming from an often misinformed populace that too often doesn't understand the workings of the police.

It's sad, too, because many of those that truly appreciate the police are those true working-class people and too few encounters with the good people does harm to how the police feel about the feelings toward them by the public. 

Still, I would say the largest number of good police do care what the public thinks of them and do their best to ensure the public still sees worth in its police. 

Hope this helps....