First of all, congratulations on being proactive and taking pride in how you do your work. Getting someone else to do the same thing is difficult if not impossible and getting it changed will take an extraordinary supervisor.
If you have computerized tracking of calls for service, you might be able to show the disparity in calls taken by you versus your coworkers.
Most of the time your coworkers and supervisors already know who is doing the work and can be depended upon to get the difficult tasks done. You will have to let them know you are not pleased with the status quo.
In St. Charles we only have presently 2 CSO's with supervision from the traffic unit Sgt and day to day control by the shift supervisor, which has worked out pretty well so far. I used to be under the control of the Operations Commander, assigned to day shift, which worked poorly for this kind of problem because the Commander was too far removed from the day to day stuff and the Sgt's pretty much ignored us as we were the Commanders responsibility.
Every place you work there will be those that do less, or have a "different philosophy" on how to do the work. The patrol officers you work with have the same problem, there are workers and there are slackers. Be proud of the fact you are a worker and, if it does not work out that the supervisor can motivate them, do not dwell on it or it will drive you crazy.
It could be worse, you could have said something to the coworker and be hauled in and told it was harassment. Can't ever open your mouth again about anything. Be careful how you approach it!

Harry,

Unfortunately, when CSO's fall under the command of Sgt's or Lt's, too many things fall through the cracks and its difficult to control when the Sgts or Lts are too occupied with the sworn patrol officers. I am a civilian CSO Supervisor in the Chicago area where for the last 25 years CSO's have been under the supervision of civilian management. I am very proud to say that all of our CSO's share duties and responsibilities equally and if the situation does arise where one may be lacking in an area, the problem is remedied immediately. The only advice I can offer to you is to do your job the best you can do it and the bad seeds will eventually be weeded out. You can't control what others do, but you can hold your head up high knowing you are doing your best!

I believe that most CSO's work as one or two per department and do not have a dedicated supervisor to help them. You have to find the supervisor that you trust, or believe has the capability to handle the situation, and approach them about the problem.
I would try to be low key about it as well. Please don't wait until you can't stand it any more. (I have done that and I can tell you it can cause more problems for you than the original complaint.)
And don't get your hopes up too high in case little changes.